bschooladmit.files.wordpress.com€¦ · Web view(2) The degree measure of angle ABC is twice the...

Bunuel DS Questions with Explanations

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!

DS Questions

1. An integer greater than 1 that is not prime is called composite. If the two-digit integer n is greater than 20, is n composite? (1) The tens digit of n is a factor of the units digit of n. (2) The tens digit of n is 2.

Given: --> two digit integer can be written as follows: --> , .

(1) tells that , ( ) --> --> as , will always be composite and factor of . Sufficient

(2) tell that , but can be for instance composite or prime . Not sufficient.

Answer: A.

2. Is the measure of one of the interior angles of quadrilateral ABCD equal to 60? (1) Two of the interior angles of ABCD are right angles. (2) The degree measure of angle ABC is twice the degree measure of angle BCD.

Sum of inner angels of quadrilateral is 360 degrees. (Sum of inner angles of polygon=180(n-2), where n is # of sides)(1) Angles can be 90+90 + any combination of two angels totaling 180. Not sufficient.(2) <ABC=2<BCD. Not sufficient

(1)+(2) Angles can be 90+90+45+135 Or 90+90+60+120 Not sufficient.

Answer: E.

3. Is x + y < 1 ? (1) x < 8/9 (2) y < 1/8

(1) Info only about x. Not sufficient(2) Info only about y. Not sufficient

All contents have been taken form www.gmatclub.com . I do not own the copyright of the any of the materials. ---ASAX

http://www.gmatclub.com/

http://gmatclub.com/forum/devil-s-dozen-129312.html#p1060761

http://gmatclub.com/forum/the-discreet-charm-of-the-ds-126962-40.html

http://gmatclub.com/forum/tough-and-tricky-exponents-and-roots-questions-125967.html

http://gmatclub.com/forum/tough-and-tricky-exponents-and-roots-questions-125967.html

http://gmatclub.com/forum/700-gmat-data-sufficiency-questions-with-explanations-100617.html

http://gmatclub.com/forum/inequality-and-absolute-value-questions-from-my-collection-86939.html

http://gmatclub.com/forum/ds-questions-about-standard-deviation-85896.html

http://gmatclub.com/forum/ds-questions-about-standard-deviation-85896.html

http://gmatclub.com/forum/new-set-of-good-ds-85441.html

http://gmatclub.com/forum/good-set-of-ds-85413.html

http://gmatclub.com/forum/set-of-8-ds-questions-85290.html


(1)+(2) x+y<73/72 Not sufficient

Answer: E.

4. Is x^4 + y^4 > z^4 ? (1) x^2 + y^2 > z^2 (2) x+y > z

Plugging numbers from Pythagorean triplets is the best way to get that not sufficient.

Answer E.

5. At a certain theater, the cost of each adult's ticket is $5 and the cost of each child's ticket is $2. What was the average cost of all the adult's and children's tickets sold at the theater yesterday? (1) Yesterday ratio of # of children's ticket sold to the # of adult's ticketr sold was 3 to 2 (2) Yesterday 80 adult's tickets were sold at the theater.

Av. cost=(2*C+5*A)/(C+A)(1) 3A=2C A=2C/3 --> Av.cost C(2+5*2/3)/C(1+2/3) --> (2+5*2/3)/(1+2/3) Sufficient(2) A=80 know nothing about C Not sufficient.

Answer: A

6. Are some goats not cows? (1) All cows are lions (2) All lions are goats.

This is good one:Question generally asks is g>c? (1) c<=l Not sufficient(2) l<=g Not sufficient(1)+(2) c<=l<=g --> If all cows are lions and all lions are goats there are no goat, which are not cows, in other case there are, so Not sufficient

Answer: E.

7. Patrick is cleaning his house in anticipation of the arrival of guests. He needs to vacuum the




floors, fold the laundry, and put away the dishes after the dishwasher completes its cycle. If the dishwasher is currently running and has 55 minutes remaining in its cycle, can Patrick complete all of the tasks before his guests arrive in exactly 1 hour? (1) Vacuuming the floors and folding the laundry will take Patrick 36 minutes. (2) Putting away the dishes will take Patrick 7 minutes.

(1) Don't know how much time is needed to put away dishes. Not sufficient(2) If dishwasher will stop after 55 min and 7 min is needed to put away dishes 55+7=62>60, so Patrick won't complete all of the tasks before his guests arrive in exactly 1 hour. Sufficient

Answer: B

8. Are all of the numbers in a certain list of 15 numbers equal? (1) The sum of all the numbers in the list is 60. (2) The sum of any 3 numbers in the list is 12.

(1) S=60 list can contain numerous combination of 15 numbers totaling 60. Not sufficient.(2) If the sum of ANY 3 numbers=12 all numbers=12/3=4. Sufficient.

Answer: B.

http://gmatclub.com/forum/collection-of-8-ds-questions-85290.html

DS Questions New:

1. The sum of n consecutive positive integers is 45. What is the value of n?(1) n is even(2) n < 9

(1) n=2 --> 22+23=45, n=4 --> n=6 x1+(x1+1)+(x1+2)+(x1+3)+(x1+4)+(x1+5)=45 x1=5. At least two options for n. Not sufficient.(2) n<9 same thing not sufficient.(1)+(2) No new info. Not sufficient.

Answer: E.

2. Is a product of three integers XYZ a prime? (1) X=-Y (2) Z=1

(1) x=-y --> for xyz to be a prime z must be -p AND x=-y shouldn't be zero. Not sufficient.



http://gmatclub.com/forum/collection-of-8-ds-questions-85290.html


(2) z=1 --> Not sufficient.(1)+(2) x=-y and z=1 --> x and y can be zero, xyz=0 not prime OR xyz is negative, so not prime. In either case we know xyz not prime.

Answer: C

3. Multiplication of the two digit numbers wx and cx, where w,x and c are unique non-zero digits, the product is a three digit number. What is w+c-x? (1) The three digits of the product are all the same and different from w c and x. (2) x and w+c are odd numbers.

(1) wx+cx=aaa (111, 222, ... 999=37*k) --> As x is the units digit in both numbers, a can be 1,4,6 or 9 (2,3,7 out because x^2 can not end with 2,3, or 7. 5 is out because in that case x also should be 5 and we know that x and a are distinct numbers).1 is also out because 111=37*3 and we need 2 two digit numbers. 444=37*12 no good we need units digit to be the same.666=37*18 no good we need units digit to be the same.999=37*27 is the only possibility all digits are distinct except the unit digits of multiples.Sufficient(2) x and w+c are odd numbers.Number of choices: 13 and 23 or 19 and 29 and w+c-x is the different even number.

Answer: A.

4. Is y – x positive?(1) y > 0(2) x = 1 – y

Easy one even if y>0 and x+y=1, we can find the x,y when y-x>0 and y-x<0Answer: E.

5. If a and b are integers, and a not= b, is |a|b > 0? (1) |a^b| > 0 (2) |a|^b is a non-zero integer

This is tricky |a|b > 0 to hold true: a#0 and b>0.

(1) |a^b|>0 only says that a#0, because only way |a^b| not to be positive is when a=0. Not sufficient. NOTE having absolute value of variable |a|, doesn't mean it's positive. It's not




negative --> |a|>=0

(2) |a|^b is a non-zero integer. What is the difference between (1) and (2)? Well this is the tricky part: (2) says that a#0 and plus to this gives us two possibilities as it states that it's integer:A. -1>a>1 (|a|>1), on this case b can be any positive integer: because if b is negative |a|^b can not be integer.ORB. |a|=1 (a=-1 or 1) and b can be any integer, positive or negative.So (2) also gives us two options for b. Not sufficient.

(1)+(2) nothing new: a#0 and two options for b depending on a. Not sufficient.

Answer: E.

6. If M and N are integers, is (10^M + N)/3 an integer?(1) N = 5(2) MN is even

Note: it's not given that M and N are positive.(1) N=5 --> if M>0 (10^M + N)/3 is an integer ((1+5)/3), if M<0 (10^M + N)/3 is a fraction ((1/10^|M|+5)/3). Not sufficient.(2) MN is even --> one of them or both positive/negative AND one of them or both even. Not sufficient(1)+(2) N=5 MN even --> still M can be negative or positive. Not sufficient.

Answer: E.

7. If b, c, and d are constants and x^2 + bx + c = (x + d)^2 for all values of x, what is the value of c? (1) d = 3 (2) b = 6

Note this part: "for all values of x"So, it must be true for x=0 --> c=d^2 --> b=2d(1) d = 3 --> c=9 Sufficient(2) b = 6 --> b=2d, d=3 --> c=9 Sufficient

Answer: D.




8. If x and y are non-zero integers and |x| + |y| = 32, what is xy? (1) -4x - 12y = 0 (2) |x| - |y| = 16

(1) x+3y=0 --> x and y have opposite signs --> either 4y=32 y=8 x=-3, xy=-24 OR -4y=32 y=-8 x=3 xy=24. The same answer. Sufficient.(2) Multiple choices. Not sufficient.

Answer: A.

9. Is the integer n odd (1) n is divisible by 3 (2) 2n is divisible by twice as many positive integers as n

(1) 3 or 6. Clearly not sufficient.(2) TIP:When odd number n is doubled, 2n has twice as many factors as n.Thats because odd number has only odd factors and when we multiply n by two we remain all these odd factors as divisors and adding exactly the same number of even divisors, which are odd*2.

Sufficient.

Answer: B.

10. The sum of n consecutive positive integers is 45. What is the value of n?(1) n is odd(2) n >= 9

Look at the Q 1 we changed even to odd and n<9 to n>=9

(1) not sufficient see Q1.(2) As we have consecutive positive integers max for n is 9: 1+2+3+...+9=45. (If n>9=10 first term must be zero. and we are given that all terms are positive) So only case n=9. Sufficient.

Answer: B.

http://gmatclub.com/forum/the-sum-of-n-consecutive-positive-integers-is-85413.html



http://gmatclub.com/forum/the-sum-of-n-consecutive-positive-integers-is-85413.html


1. When the positive integer x is divided by 4, is the remainder equal to 3? (1) When x/3 is divided by 2, the remainder is 1. (2) x is divisible by 5.

Answer: E.

2. In 2003 Acme Computer priced its computers five times higher than its printers. What is the ratio of its gross revenue for computers and printers respectively in the year 2003? (1) In the first half of 2003 it sold computers and printers in the ratio of 3:2, respectively, and in the second half in the ratio of 2:1. (2) It sold each computer for $1000.

Answer: E

3. Last Tuesday a trucker paid $155.76, including 10 percent state and federal taxes, for diesel fuel. What was the price per gallon for the fuel if the taxes are excluded? (1) The trucker paid $0.118 per gallon in state and federal taxes on the fuel last Tuesday. (2) The trucker purchased 120 gallons of the fuel last Tuesday.

Answer: D.

4. What is the remainder when the positive integer x is divided by 8? (1) When x is divided by 12, the remainder is 5. (2) When x is divided by 18, the remainder is 11.

Answer: E.

5. Al, Pablo, and Marsha shared the driving on a 1500 mile trip, which of the three drove the greatest distance on the trip? (1) Al drove 1 hour longer than Pablo but at an average of 5 miles per hour slower than Pablo.(2) Marsha drove 9 hours and averaged 50 miles per hour

Answer: E.




6. How many perfect squares are less than the integer d? (1) 23 < d < 33 (2) 27 < d < 37

Answer: B.

7. The integers m and p are such that 2 is less than m and m is less than p. Also, m is not a factor of p. If r is the remainder when p is divided by m, is r > 1. (1) The greatest common factor of m and p is 2. (2) The least common multiple of m and p is 30.

Answer: A.

8. A scientist is studying bacteria whose cell population doubles at constant intervals, at which times each cell in the population divides simultaneously. Four hours from now, immediately after the population doubles, the scientist will destroy the entire sample. How many cells will the population contain when the bacteria is destroyed? (1) Since the population divided two hours ago, the population has quadrupled, increasing by 3,750 cells. (2) The population will double to 40,000 cells with one hour remaining until the scientist destroys the sample.

Answer: C.

9. Is x^2 equal to xy? (1) x^2 - y^2 = (x+5)(y-5) (2) x=y

Answer: B.

10. A number of oranges are to be distributed evenly among a number of baskets. Each basket will contain at least one orange. If there are 20 oranges to be distributed, what is the number of oranges per basket? (1) If the number of baskets were halved and all other conditions remained the same, there would be twice as many oranges in every remaining basket. (2) If the number of baskets were doubled, it would no longer be possible to place at least one orange in every basket.




Answer: B.

11. If p is a prime number greater than 2, what is the value of p?(1) There are a total of 100 prime numbers between 1 and p+1(2) There are a total of p prime numbers between 1 and 3912.

Answer: D.

12. If x is a positive integer, what is the least common multiple of x, 6, and 9?(1) The LCM of x and 6 is 30.(2) The LCM of x and 9 is 45.

Answer: D.

http://gmatclub.com/forum/collection-of-12-ds-questions-85441-20.html#p642315

Standard Deviation:

Please note the following: A. I was assured MANY TIMS, by various GMAT tutors, that GMAT won't ask you to actually calculate SD, but rather to understand the concept of it. Though KNOWING how it's calculated helps in understanding the concept.B. During the real GMAT it's highly unlikely to get more than one ot two question on SD (as on combinatorics), actually you may see none, so do not spend too much of your preparation time on it, it's better to concentrate on issues you'll definitely face on G-day.

Many questions below are easy, some are tough, but anyway they are good to master in solving SD problems. I'll post OA after some discussions. Please provide your way of thinking along with the answer. Thanks.

1. What is the standard deviation of Company R’s earnings per month for this year? (1) The standard deviation of Company R’s earnings per month in the first half of this year was $2.3 million. (2) The standard deviation of Company R’s earnings per month in the second half of this year was $3.9 million. Answer: E.

2. What is the standard deviation of Q, a set of consecutive integers? (1) Q has 21 members. (2) The median value of set Q is 20. Answer: A.



http://gmatclub.com/forum/collection-of-12-ds-questions-85441-20.html#p642315


3. Lifetime of all the batteries produced by certain companies have a distribution which is symmetric about mean m. If the distribution has a standard deviation of d , what percentage of distribution is greater than m+d? (1) 68 % of the distribution in the interval from m-d to m+d, inclusive (2) 16% of the distribution is less than m-d Answer: D.

4. Question deleted

5. List S and list T each contain 5 positive integers, and for each list the average of the integers in the list is 40. If the integers 30,40 and 50 are in both lists , is the standard deviation of the integers in list S greater than the standard deviation of the integers in list T?(1)The integer 25 is in list S (2)The integer 45 is in list T Answer: C.

6. Set T consists of odd integers divisible by 5. Is standard deviation of T positive? (1) All members of T are positive (2) T consists of only one memberAnswer: B.

7. Set X consists of 8 integers. Is the standard deviation of set X equal to zero? (1) The range of set X is equal to 3 (2) The mean of set X is equal to 5 Answer: A.

8. {x,y,z}If the first term in the data set above is 3, what is the third term? (1) The range of this data set is 0. (2) The standard deviation of this data set is 0.Answer: D.

9. Question deleted

10. A scientist recorded the number of eggs in each of 10 birds' nests. What was the standard deviation of the numbers of eggs in the 10 nests? (1) The average (arithmetic mean) number of eggs for the 10 nests was 4. (2) Each of the 10 nests contained the same number of eggs.Answer: B.




11. During an experiment, some water was removed from each of the 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment? (1) For each tank, 30% of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment. (2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons. Answer: A.

http://gmatclub.com/forum/lately-many-questions-were-asked-about-the-standard-85896.html

Inequality and absolute value.

1. If 6*x*y = x^2*y + 9*y, what is the value of xy? (1) y – x = 3(2) x^3< 0

First let's simplify given expression :

--> . Note here that we CAN NOT reduce this expression by , as some of you did. Remember we are asked to determine the value of

, and when reducing by you are assuming that doesn't equal to . We don't know that.

Next: we can conclude that either or/and . Which means that equals to 0,

when y=0 and x any value (including 3), OR when y is not equal to zero, and x=3.

(1) . If y is not 0, x must be 3 and y-x to be 3, y must be 6. In this case . But if y=0 then x=-3 and . Two possible scenarios. Not sufficient.

OR:

--> --> --> either or --> if , then and if , then and

. Two different answers. Not sufficient.

(2) . x is negative, hence x is not equals to 3, hence y must be 0. So, xy=0. Sufficient.

Answer: B.






2. 2. If y is an integer and y = |x| + x, is y = 0? (1) x < 0 (2) y < 1

Note: as then is never negative. For then and for then (when x is negative or zero) then .

(1) --> . Sufficient.

(2) , as we concluded y is never negative, and we are given that is an integer, hence . Sufficient.

Answer: D.

3. Is x^2 + y^2 > 4a?(1) (x + y)^2 = 9a (2) (x – y)^2 = a

(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x,y, and a could be 0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not sufficient.

Answer: E.

4. 4. Are x and y both positive? (1) 2x-2y=1(2) x/y>1

(1) 2x-2y=1. Well this one is clearly insufficient. You can do it with number plugging OR consider the following: x and y both positive means that point (x,y) is in the I quadrant. 2x-2y=1 --> y=x-1/2, we know it's an equation of a line and basically question asks whether this line (all (x,y) points of this line) is only in I quadrant. It's just not possible. Not sufficient.

(2) x/y>1 --> x and y have the same sign. But we don't know whether they are both positive or both negative. Not sufficient.

(1)+(2) Again it can be done with different approaches. You should just find the one which is the less time-consuming and comfortable for you personally.




One of the approaches:

-->

--> --> substitute x --> --> is positive, and as , is positive too. Sufficient.

Answer: C.

5. 5. What is the value of y?(1) 3|x^2 -4| = y - 2 (2) |3 - y| = 11

(1) As we are asked to find the value of y, from this statement we can conclude only that y>=2, as LHS is absolute value which is never negative, hence RHS als can not be negative. Not sufficient.

(2) |3 - y| = 11:

y<3 --> 3-y=11 --> y=-8y>=3 --> -3+y=11 --> y=14

Two values for y. Not sufficient.

(1)+(2) y>=2, hence y=14. Sufficient.

Answer: C.

6. 6. If x and y are integer, is y > 0? (1) x +1 > 0 (2) xy > 0

(1) x+1>0 --> x>-1. As x is an integer x can take the following values 0,1,2,... But we know nothing about y. Not sufficient.

(2) xy>0. x and y have the same sign (both positive OR both negative) and neither x nor y is zero. Not sufficient.

(1)+(2) x is positive, as from (1) it's 0,1,2.. and from (2) x is not zero. Hence xy to be positive y also must be positive. Sufficient.




Answer: C. ____________

7. 7. |x+2|=|y+2| what is the value of x+y?(1) xy<0(2) x>2 y<2

This one is quite interesting.

First note that |x+2|=|y+2| can take only two possible forms:

A. x+2=y+2 --> x=y. This will occur if and only x and y are both >= than -2 OR both <= than -2. In that case x=y. Which means that their product will always be positive or zero when x=y=0.B. x+2=-y-2 --> x+y=-4. This will occur when either x or y is less then -2 and the other is more than -2.

When we have scenario A, xy will be nonnegative only. Hence if xy is negative we have scenario B and x+y=-4. Also note that vise-versa is not right. Meaning that we can have scenario B and xy may be positive as well as negative.

(1) xy<0 --> We have scenario B, hence x+y=-4. Sufficient.

(2) x>2 and y<2, x is not equal to y, we don't have scenario A, hence we have scenario B, hence x+y=-4. Sufficient.

Answer: D.

8. 8. a*b#0. Is |a|/|b|=a/b?(1) |a*b|=a*b(2) |a|/|b|=|a/b|

|a|/|b|=a/b is true if and only a and b have the same sign, meaning a/b is positive.

(1) |a*b|=a*b, means a and b are both positive or both negative, as LHS is never negative (well in this case LHS is positive as neither a nor b equals to zero). Hence a/b is positive in any case. Hence |a|/|b|=a/b. Sufficient.

(2) |a|/|b|=|a/b|, from this we can not conclude whether they have the same sign or not. Not sufficient.

Answer: A.




9. 9. Is n<0?(1) -n=|-n| (2) n^2=16

(1) -n=|-n|, means that either n is negative OR n equals to zero. We are asked whether n is negative so we can not be sure. Not sufficient.

(2) n^2=16 --> n=4 or n=-4. Not sufficient.

(1)+(2) n is negative OR n equals to zero from (1), n is 4 or -4 from (2). --> n=-4, hence it's negative, sufficient.

Answer: C.

10. If n is not equal to 0, is |n| < 4 ? (1) n^2 > 16(2) 1/|n| > n

Question basically asks is -4<n<4 true.

(1) n^2>16 --> n>4 or n<-4, the answer to the question is NO. Sufficient.

(2) 1/|n| > n, this is true for all negative values of n, hence we can not answer the question. Not sufficient.

Answer: A.11. Is |x+y|>|x-y|? (1) |x| > |y|(2) |x-y| < |x|

To answer this question you should visualize it. We have comparison of two absolute values. Ask yourself when |x+y| is more then than |x-y|? If and only when x and y have the same sign absolute value of x+y will always be more than absolute value of x-y. As x+y when they have the same sign will contribute to each other and x-y will not.

5+3=8 and 5-3=2 OR -5-3=-8 and -5-(-3)=-2.

So if we could somehow conclude that x and y have the same sign or not we would be able to answer the question.

(1) |x| > |y|, this tell us nothing about the signs of x and y. Not sufficient.




(2) |x-y| < |x|, says that the distance between x and y is less than distance between x and origin. This can only happen when x and y have the same sign, when they are both positive or both negative, when they are at the same side from the origin. Sufficient. (Note that vise-versa is not right, meaning that x and y can have the same sign but |x| can be less than |x-y|, but if |x|>|x-y| the only possibility is x and y to have the same sign.)

Answer: B.

11. 12. Is r=s?(1) -s<=r<=s (2) |r|>=s

This one is tough.

(1) -s<=r<=s, we can conclude two things from this statement:A. s is either positive or zero, as -s<=s;B. r is in the range (-s,s) inclusive, meaning that r can be -s as well as s.But we don't know whether r=s or not. Not sufficient.

(2) |r|>=s, clearly insufficient.

(1)+(2) -s<=r<=s, s is not negative, |r|>=s --> r>=s or r<=-s. This doesn't imply that r=s, from this r can be -s as well. Consider: s=5, r=5 --> -5<=5<=5 |5|>=5s=5, r=-5 --> -5<=-5<=5 |-5|>=5Both statements are true with these values. Hence insufficient.

Answer: E. _________________

12. 13. Is |x-1| < 1?(1) (x-1)^2 <= 1 (2) x^2 - 1 > 0

Last one.

Is |x-1| < 1? Basically the question asks is 0<x<2 true?

(1) (x-1)^2 <= 1 --> x^2-2x<=0 --> x(x-2)<=0 --> 0<=x<=2. x is in the range (0,2) inclusive. This is the trick here. x can be 0 or 2! Else it would be sufficient. So not sufficient.




(2) x^2 - 1 > 0 --> x<-1 or x>1. Not sufficient.

(1)+(2) Intersection of the ranges from 1 and 2 is 1<x<=2. Again 2 is included in the range, thus as x can be 2, we can not say for sure that 0<x<2 is true. Not sufficient.

Answer: E.

http://gmatclub.com/forum/inequality-and-absolute-value-questions-from-my-collection-86939-40.html

Exponents and Roots:

1. If , where and are positive integers, what is the units digit of ?

(1)

(2)

(1) --> since both and are positive integers then and are perfect squares --> there are only two perfect squares in the given range 121=11^2 and 144=12^2 --> and . Sufficient.(As cyclicity of units digit of in integer

power is , therefore the units digit of is the same as the units digit of , so 3).

(2) --> --> since both and are positive integers then: and --> and . Sufficient.

Answer: D.

2. 2. If x, y, and z are positive integers and . Is and integer?(1) is an even perfect square and is an odd perfect cube.(2) is not an integer.

Note: a perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is a perfect square. Similarly a perfect cube, is an integer that can be written as the cube of some other integer. For example 27=3^3, is a perfect cube.

Make prime factorization of 2,700 --> .

(1) is an even perfect square and is an odd perfect cube --> if is either or

and then must be a perfect square

which makes an integer: or . But if






then could be which makes not an integer. Not sufficient.

(2) is not an integer. Clearly insufficient.

(1)+(2) As from (1) then , therefore it must be (from 1) --> is a perfect square which makes an integer: or . Sufficient.

Answer: C.

3. 3. If then what is the value of ?(1) (2)

--> factor out from the nominator and apply to

the expression in the denominator: . So we should find the value of .

(1) --> --> --> (note that since then the second solution is not valid). Sufficient.

(2) . Not sufficient.

Answer: A.

4. 4. If is ?

(1)

(2)

means that neither of unknown is equal to zero. Next,

, so the question becomes: is ? Since

and are positive numbers then the question boils down whether , which is the same as whether (recall that odd roots have the same sign as the base of the

root, for example: and ).




(1) --> as even root from positive number ( in our case) is positive then

, (or which is the same ). Therefore answer to the original question is NO. Sufficient.

(2) --> the same here as then , (or which is the same ). Therefore answer to the original question is NO. Sufficient.

Answer: D.

5. 5. If and are negative integers, then what is the value of ?

(1)

(2)

(1) --> as both and are negative integers then

--> or . Note that as negative integer (x) in negative integer power (y) gives positive number (1/81) then the power must be negative even number. Not sufficient.

(2) --> as the result is negative then must be negative odd number -->

--> or . Not sufficient.

(1)+(2) Only one pair of negative integers and satisfies both statements and --> . Sufficient.

Answer: C.

6. 6. If then what is the value of ?

(1)

(2) and

(1) --> --> applying

we'll get: --> -->

--> , since given that then hence .




Sufficient.

(2) and --> since and then the only case to hold true is

when --> . Sufficient.

Answer: D.

7. 7. If is a positive integer is an integer?

(1) is an integer (2) is not an integer

Must know for the GMAT: if is a positive integer then is either a positive integer itself or an irrational number. (It can not be some reduced fraction eg 7/3 or 1/2)

Also note that the question basically asks whether is a perfect square.

(1) is an integer --> can not be a perfect square because if it is, for example if

for some positive integer then . Sufficient.

(2) is not an integer --> --> . Sufficient.

Answer: D.

8. 8. What is the value of ?

(1)

(2)

(1) --> ---> --> . Sufficient.

(2) --> --> --> -->

(the power of 3 must be zero in order this equation to hold true) -->

the sum of two non-negative values is zero --> both and must be

zero --> --> . Sufficient.

Answer: D.




9. 9. If , and are non-zero numbers, what is the value of ?(1) (2)

(1) --> infinitely many combinations of , and are possible which will give different values of the expression in the stem: try x=y=1 and y=-6 or x=1, y=2, z=-3. Not sufficient.

(2) --> --> substitute this value of x into the expression in

the stem --> , as then: . Sufficient.

Must know for the GMAT:

and

.

Answer: B.

10. 10. If and are non-negative integers and is an even integer?

(1)

(2)

(1) --> --> equate the powers:

--> .

Since both and are integers (and ) then and -->

and --> , so the answer to the question is No. Sufficient. (Note that and --> and is not a valid scenario (solution) as both unknowns must be integers)

(2) --> obviously must be an integer (since

) and as then the only solution is

--> . So, --> two scenarios are possible:




A. and (notice that holds true) --> , and

in this case: ;

B. and (notice that holds true) --> , and

in this case: .

Two different answers. Not sufficient.

Answer: A.

11. 11. What is the value of ?(1)

(2)

Notice that we are not told that the and are integers.

(1) --> if and are integers then as then and BUT if they are not, then for any value of there will exist some non-integer

to satisfy given expression and vise-versa (for example if then --> --> ). Not

sufficient.

(2) --> --> either and is ANY number (including 2) or and is ANY number (including 1). Not sufficient.

(1)+(2) If from (2) then from (1) --> and if from (2) then from (1) --> . Thus and .

Sufficient.

Answer: C.

http://gmatclub.com/forum/tough-and-tricky-exponents-and-roots-questions-125967-20.html

Discreet Chart of DS:

1. Bonnie can paint a stolen car in x hours, and Clyde can paint the same car in y hours. They start working simultaneously and independently at their respective constant rates at 9:45am. If both x and y are odd integers, is x=y?(1) x^2+y^2<12(2) Bonnie and Clyde complete the painting of the car at 10:30am






Solution: the-discreet-charm-of-the-ds-126962-20.html#p1039633

2. Is xy<=1/2? (1) x^2+y^2=1(2) x^2-y^2=0


3. If a, b and c are integers, is abc an even integer?(1) b is halfway between a and c(2) a = b - c


4. How many numbers of 5 consecutive positive integers is divisible by 4?(1) The median of these numbers is odd(2) The average (arithmetic mean) of these numbers is a prime number


5. What is the value of integer x?(1) 2x^2+9<9x(2) |x+10|=2x+8


6. If a and b are integers and ab=2, is a=2?(1) b+3 is not a prime number(2) a>b


7. A certain fruit stand sold total of 76 oranges to 19 customers. How many of them bought only one orange?(1) None of the customers bought more than 4 oranges (2) The difference between the number of oranges bought by any two customers is even


8. If x=0.abcd, where a, b, c and d are digits from 0 to 9, inclusive, is x>7/9?(1) a+b>14



http://gmatclub.com/forum/the-discreet-charm-of-the-ds-126962-40.html#p1039655








(2) a-c>6


9. If x and y are negative numbers, is x<y?(1) 3x + 4 < 2y + 3(2) 2x - 3 < 3y - 4


10. The function f is defined for all positive integers a and b by the following rule: f(a,b)=(a+b)/GCF(a,b), where GCF(a,b) is the greatest common factor of a and b. If f(10,x)=11, what is the value of x?(1) x is a square of an integer(2) The sum of the distinct prime factors of x is a prime number.


11. If x and y are integers, is x a positive integer?(1) x*|y| is a prime number.(2) x*|y| is non-negative integer.


12. If 6a=3b=7c, what is the value of a+b+c?(1) ac=6b(2) 5b=8a+4c


Discreet Charm of DS Solutions by Bunuel:

1. Bonnie can paint a stolen car in x hours, and Clyde can paint the same car in y hours. They start working simultaneously and independently at their respective constant rates at 9:45am. If both x and y are odd integers, is x=y?

Bonnie and Clyde when working together complete the painting of the car ins hours (sum of the rates equal to the combined rate or reciprocal of total time:

--> ). Now, if then the total time would be: ,









since is odd then this time would be odd/2: 0.5 hours, 1.5 hours, 2.5 hours, ....

(1) x^2+y^2<12 --> it's possible and to be odd and equal to each other if but it's also possible that and (or vise-versa). Not

sufficient.

(2) Bonnie and Clyde complete the painting of the car at 10:30am --> they complete the job in 3/4 of an hour (45 minutes), since it's not odd/2 then and are not equal. Sufficient.

Answer: B.

2. 2. Is xy<=1/2?

(1) x^2+y^2=1. Recall that (square of any number is more than or

equal to zero) --> --> since then: -->

. Sufficient.

(2) x^2-y^2=0 --> . Clearly insufficient.

Answer: A.

3. 3. If a, b and c are integers, is abc an even integer?

In order the product of the integers to be even at leas on of them must be even

(1) b is halfway between a and c --> on the GMAT we often see such statement and

it can ALWAYS be expressed algebraically as . Now, does that mean that at leas on of them is be even? Not necessarily: , and , of course it's also possible that for example , for and . Not sufficient.

(2) a = b - c --> . Since it's not possible that the sum of two odd integers to be odd then the case of 3 odd numbers is ruled out, hence at least on of them must be even. Sufficient.

Answer: B.

4. 4. How many numbers of 5 consecutive positive integers is divisible by 4?

(1) The median of these numbers is odd --> the median of the set with odd number




of terms is just a middle term, thus our set of 5 consecutive numbers is: {Odd, Even, Odd, Even, Odd}. Out of 2 consecutive even integers only one is a multiple of 4. Sufficient.

(2) The average (arithmetic mean) of these numbers is a prime number --> in any evenly spaced set the arithmetic mean (average) is equal to the median --> mean=median=prime. Since it's not possible that median=2=even, (in this case not all 5 numbers will be positive), then median=odd prime, and we have the same case as above. Sufficient.

Answer: D.

5. 5. What is the value of integer x?

(1) 2x^2+9<9x --> factor qudratics: --> roots are and 3 --> "<" sign indicates that the solution lies between the roots: --> since there only integer in this range is 2 then . Sufficient.

(2) |x+10|=2x+8 --> LHS is an absolute value, which is always non negative, hence RHS must also be non-negative: --> , for this range is

positive hence --> --> . Sufficient.

Answer: D.

Check this for more on solving inequalities like the one in the first statement:x2-4x-94661.html#p731476 inequalities-trick-91482.htmleverything-is-less-than-zero-108884.html?hilit=extreme#p868863xy-plane-71492.html?hilit=solving%20quadratic#p841486

6. 6. If a and b are integers and ab=2, is a=2?

Notice that we are not told that a and b are positive.

There are following integer pairs of (a, b) possible: (1, 2), (-1, -2), (2, 1) and (-2, -1). Basically we are asked whether we have the third case.

(1) b+3 is not a prime number --> rules out 1st and 4th options. Not sufficient.(2) a>b --> again rules out 1st and 4th options. Not sufficient.

(1)+(2) Still two options are left: (-1, -2) and (2, 1). Not sufficient.



http://gmatclub.com/forum/xy-plane-71492.html?hilit=solving%20quadratic#p841486

http://gmatclub.com/forum/everything-is-less-than-zero-108884.html?hilit=extreme#p868863

http://gmatclub.com/forum/inequalities-trick-91482.html

http://gmatclub.com/forum/x2-4x-94661.html#p731476


Answer: E.

7. 7. A certain fruit stand sold total of 76 oranges to 19 customers. How many of them bought only one orange?

(1) None of the customers bought more than 4 oranges --> this basically means that all customers bought exactly 4 oranges (76/19=4), because if even one customer bought less than 4, the sum will be less than 76. Hence, no one bought only one orange. Sufficient.

(2) The difference between the number of oranges bought by any two customers is even --> in order the difference between ANY number of oranges bought to be even, either all customers must have bought odd number of oranges or all customers must have bough even number of oranges. But the first case is not possible: the sum of 19 odd numbers is odd and not even like 76. Hence, again no one bought only one=odd orange. Sufficient.

Answer: D.

8. 8. If x=0.abcd, where a, b, c and d are digits from 0 to 9, inclusive, is x>7/9?

First of all 7/9 is a recurring decimal =0.77(7). For more on converting Converting Decimals to Fractions see: math-number-theory-88376.html

(1) a+b>14 --> the least value of a is 6 (6+9=15>14), so in this case x=0.69d<0.77(7) but a=7 and b=9 is also possible, and in this case x=0.79d>0.77(7). Not sufficient.

(2) a-c>6 --> the least value of a is 7 (7-0=7>6), but we don't know the value of b. Not sufficient.

(1)+(2) The least value of a is 7 and in this case from (1) least value of b is 8 (7+8=15>14), hence the least value of x=0.78d>0.77(7). Sufficient.

Answer: C. ___________

9. 9. If x and y are negative numbers, is x<y?

(1) 3x + 4 < 2y + 3 --> . can be some very small number for instance -100 and some large enough number for instance -3 and the answer would be YES,

BUT if and then the answer would be NO, . Not



http://gmatclub.com/forum/math-number-theory-88376.html


sufficient.

(2) 2x - 3 < 3y - 4 --> --> --> (as y+negative is "more negative" than y). Sufficient.

Answer: B.

10. 10. The function f is defined for all positive integers a and b by the following rule: f(a,b)=(a+b)/GCF(a,b), where GCF(a,b) is the greatest common factor of a and b. If f(10,x)=11, what is the value of x?

Notice that the greatest common factor of 10 and x, GCF(10,x), naturally must be a factor of 10: 1, 2, 5, and 10. Thus from f(10,x)=11 we can get four different values of x:

GCF(10,x)=1 --> --> ;

GCF(10,x)=2 --> --> ;

GCF(10,x)=5 --> --> ;

GCF(10,x)=10 --> --> .

(1) x is a square of an integer --> can be 1 or 100. Not sufficient.

(2) The sum of the distinct prime factors of x is a prime number ---> distinct primes

of 12 are 2 and 3: , distinct primes of 45 are 3 and 5:

and distinct primes of 100 are 2 and 5: . can be 12 or 100. Not sufficient.

(1)+(2) can only be 100. Sufficient.

Answer: C. _________________

11. 11. If x and y are integers, is x a positive integer?

(1) x*|y| is a prime number --> since only positive numbers can be primes, then: x*|y|=positive --> x=positive. Sufficient

(2) x*|y| is non-negative integer. Notice that we are told that x*|y| is non-negative,




not that it's positive, so x can be positive as well as zero. Not sufficient.

Answer: A.

12. 12. If 6a=3b=7c, what is the value of a+b+c?

Given: --> least common multiple of 6, 3, and 7 is 42 hence we ca write: , for some number --> , and

.

(1) ac=6b --> --> --> or . Not sufficient.

(2) 5b=8a+4c --> --> --> --> --> --> . Sufficient.

Answer: B.


Devil’s Dozen Questions

1. Jules and Jim both invested certain amount of money in bond M for one year, which pays for 12% simple interest annually. If no other investment were made, then Jules initial investment in bond M was how many dollars more than Jim's investment in bond M.(1) In one year Jules earned $24 more than Jim from bond M.(2) If the interest were 20% then in one year Jules would have earned $40 more than Jim from bond M.

Solution: devil-s-dozen-129312.html#p1063846

2. If n is a positive integer and p is a prime number, is p a factor of n!?(1) p is a factor of (n+2)!-n!(2) p is a factor of (n+2)!/n!


3. If x and y are integers, is y an even integer?(1) 4y^2+3x^2=x^4+y^4(2) y=4-x^2









4. Of the 58 patients of Vertigo Hospital, 45 have arachnophobia. How many of the patients have acrophobia?(1) The number of patients of Vertigo Hospital who have both arachnophobia and acrophobia is the same as the number of patients who have neither arachnophobia nor acrophobia.(2) 32 patients of Vertigo Hospital have arachnophobia but not acrophobia.


5. If at least one astronaut do NOT listen to Bach at Solaris space station, then how many of 35 astronauts at Solaris space station listen to Bach?(1) Of the astronauts who do NOT listen to Bach 56% are male.(2) Of the astronauts who listen to Bach 70% are female.


6. Is the perimeter of triangle with the sides a, b and c greater than 30?(1) a-b=15.(2) The area of the triangle is 50.


7. Set A consists of k distinct numbers. If n numbers are selected from the set one-by-one, where n<=k, what is the probability that numbers will be selected in ascending order?(1) Set A consists of 12 even consecutive integers.(2) n=5.

Solution: devil-s-dozen-129312-20.html#p1063874

8. If p is a positive integer, what is the remainder when p^2 is divided by 12?(1) p is greater than 3.(2) p is a prime.


9. The product of three distinct positive integers is equal to the square of the largest of the three numbers, what is the product of the two smaller numbers?(1) The average (arithmetic mean) of the three numbers is 34/3.(2) The largest number of the three distinct numbers is 24.




http://gmatclub.com/forum/devil-s-dozen-129312-20.html#p1063886







10. There is at least one viper and at least one cobra in Pandora's box. How many cobras are there?(1) There are total 99 snakes in Pandora's box. (2) From any two snakes from Pandora's box at least one is a viper.


11. Alice has $15, which is enough to buy 11 muffins and 7 brownies, is $45 enough to buy 27 muffins and 27 brownies?(1) $15 is enough to buy 7 muffins and 11 brownies.(2) $15 is enough to buy 10 muffins and 8 brownies.


12. If x>0 and xy=z, what is the value of yz?

(1) .

(2) .


13. Buster leaves the trailer at noon and walks towards the studio at a constant rate of B miles per hour. 20 minutes later, Charlie leaves the same studio and walks towards the same trailer at a constant rate of C miles per hour along the same route as Buster. Will Buster be closer to the trailer than to the studio when he passes Charlie?(1) Charlie gets to the trailer in 55 minutes. (2) Buster gets to the studio at the same time as Charlie gets to the trailer.


Devil’s Dozen Solutions by Bunuel:

1. Jules and Jim both invested certain amount of money in bond M for one year, which pays for 12% simple interest annually. If no other investment were made, then Jules initial investment in bond M was how many dollars more than Jim's investment in bond M.

Question:








(1) In one year Jules earned $24 more than Jim from bond M.

--> --> . Sufficient.

(2) If the interest were 20% then in one year Jules would have earned $40 more than Jim from bond M. Basically the same type of information as above:

--> --> . Sufficient.

Answer: D.

2. 2. If n is a positive integer and p is a prime number, is p a factor of n!?

(1) p is a factor of (n+2)!-n! --> if then and for then answer will be YES but for the answer will be NO. Not sufficient.

(2) p is a factor of (n+2)!/n! --> --> if then

and for then answer will be YES but for the answer will be NO. Not sufficient.

(1)+(2) . Now, and

are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1. So, as

from (2) is a factor of then it can not be a factor of

, thus in order to be a factor of , from (1), then it should be a factor of the first multiple of this expression: . Sufficient.

Answer: C.

3. 3. If x and y are integers, is y an even integer?

(1) 4y^2+3x^2=x^4+y^4 --> rearrange: -->

. Notice that LHS is even for any value of : if is odd then and if is even then the product is

naturally even. So, is also even, but in order it to be even must be even, since if is odd then

. Sufficient.




(2) y=4-x^2 --> if then but if then . Not sufficient.

Answer: A.

4.

5. 5. If at least one astronaut do NOT listen to Bach at Solaris space station, then how many of 35 astronauts at Solaris space station listen to Bach?

Also very tricky.

(1) Of the astronauts who do NOT listen to Bach 56% are male --> if # of astronauts who do NOT listen to Bach is then is # of males who do

NOT listen to Bach. Notice that must be an integer. Hence x must be a multiple of 25: 25, 50, 75, ... But (# of astronauts who do NOT listen to Bach) must also be less than (or equal to) 35. So can only be 25, which makes # of astronauts who do listen to Bach equal to 35-25=10. Sufficient.

(2) Of the astronauts who listen to Bach 70% are female. Now, if we apply the same logic here we get that, if # of astronauts who listen to Bach is then

is # of females who listen to Bach: must be an integer. Hence it must be a multiple of 10, but in this case it can take more than 1 value: 10, 20, 30. So, this statement is not sufficient.

Answer: A.




6. 6. Is the perimeter of triangle with the sides a, b and c greater than 30?

700+ question.

(1) a-b=15. Must know for the GMAT: the length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides. So, a+b>c>15 --> a+b+c>30. Sufficient.

(2) The area of the triangle is 50. For a given perimeter equilateral triangle has the largest area. Now, if the perimeter were equal to 30 then it would have the largest area if it were equilateral. Let's find what this area would be:

. Since even equilateral triangle with perimeter of 30 can not produce the area of 50, then the perimeter must be more that 30. Sufficient.

Answer: D.

7. 7. Set A consists of k distinct numbers. If n numbers are selected from the set one-by-one, where n<=k, what is the probability that numbers will be selected in ascending order?

(1) Set A consists of 12 even consecutive integers;(2) n=5.

We should understand following two things:1. The probability of selecting any n numbers from the set is the same. Why should any subset of n numbers have higher or lower probability of being selected than some other subset of n numbers? Probability doesn't favor any particular subset.

2. Now, consider that the subset selected is , where

. We can select this subset of numbers in # of ways and out of these n! ways only one, namely will be in ascending

order. So 1 out of n!. .

Hence, according to the above the only thing we need to know to answer the question is the size of the subset (n) we are selecting from set A.

Answer: B.




8. 8. If p is a positive integer, what is the remainder when p^2 is divided by 12?

(1) p is greater than 3.(2) p is a prime.

(1) p is greater than 3. Clearly insufficient: different values of will give different values of the remainder.(2) p is a prime. Also insufficient: if then the remainder is 4 but if then the remainder is 9.

(1)+(2) You can proceed with number plugging and try several prime numbers greater than 3 to see that the remainder will always be 1 (for example try

, , ).

If you want to double-check this with algebra you should apply the following property of the prime number: any prime number greater than 3 can be expressed either as or .

If then which gives remainder 1 when divided by 12;

If then which also gives remainder 1 when divided by 12.

Answer: C.

9. 9. The product of three distinct positive integers is equal to the square of the largest of the three numbers, what is the product of the two smaller numbers?

700 question.

Let the three integers be , , and , where . Given: --> . Question:

(1) The average (arithmetic mean) of the three numbers is 34/3 -->

--> --> . Now, since and are integers, then and . and --> . Sufficient. (Notice that and is not possible since in this case

and we are told that all integers are positive).

(2) The largest number of the three distinct numbers is 24. Directly give the




value of c. Sufficient.

Answer: D.

10. 10. There is at least one viper and at least one cobra in Pandora's box. How many cobras are there?

Quite tricky.

(1) There are total 99 snakes in Pandora's box. Clearly insufficient.

(2) From any two snakes from Pandora's box at least one is a viper. Since from ANY two snakes one is a viper then there can not be 2 (or more) cobras and since there is at least one cobra then there must be exactly one cobra in the box. Sufficient.

Answer: B.

11. 11. Alice has $15, which is enough to buy 11 muffins and 7 brownies, is $45 enough to buy 27 muffins and 27 brownies?

700+ question.

Given: , where and are prices of one muffin and one brownie respectively. Question: is ? --> . Question basically asks whether we can substitute 2 muffins with 2 brownies.

Now if we can easily substitute 2 muffins with 2 brownies (since will be more than ). But if we won't know this for sure.

But consider the case when we are told that we can substitute 3 muffins with 3 brownies. In both cases ( or ) it would mean that we can substitute 2 (so less than 3) muffins with 2 brownies, but again we won't be sure whether we can substitute 4 (so more than 3) muffins with 4 brownies.

(1) $15 is enough to buy 7 muffins and 11 brownies --> : we can substitute 4 muffins with 4 brownies, so according to above we can surely substitute 2 muffins with 2 brownies. Sufficient.(1) $15 is enough to buy 10 muffins and 8 brownies --> : we can substitute 1 muffin with 1 brownie, so according to above this is does not ensure that we can substitute 2 muffins with 2 brownies. Not sufficient.




Answer: A.

12. 12. If x>0 and xy=z, what is the value of yz?

(1) . If then and but if then ,

and . Not sufficient.

(2) --> --> --> since then:

. Sufficient.

Answer: B.

13. 13. Buster leaves the trailer at noon and walks towards the studio at a constant rate of B miles per hour. 20 minutes later, Charlie leaves the same studio and walks towards the same trailer at a constant rate of C miles per hour along the same route as Buster. Will Buster be closer to the trailer than to the studio when he passes Charlie?

(1) Charlie gets to the trailer in 55 minutes. No info about Buster. Not sufficient.

(2) Buster gets to the studio at the same time as Charlie gets to the trailer --> Charlie needed 20 minutes less than Buster to cover the same distance, which means that the rate of Charlie is higher than that of Buster. Since after they pass each other they need the same time to get to their respective destinations (they get at the same time to their respective destinations) then Buster had less distance to cover ahead (at lower rate) than he had already covered (which would be covered by Charlie at higher rate). Sufficient.

Answer: B.

1.If r is the remainder when the positive integer n is divided by 7, what is the value of r

1. when n is divided by 21, the remainder is an odd number2. when n is divided by 28, the remainder is 3

The possible reminders can be 1,2,3,4,5 and 6. We have the pinpoint the exact remainder from this 6 numbers.

St 1: when n is divided by 21 ( 7 and 3) the remainder is an odd number. But it cannot be 7, 3 or 9 . Hence the possibilities are : 1 and 5.




Hence there can be two remainders ,1 and 5, when divided by 7.NOT SUFFICIENT

St 2: when n is divided by 28 the remainder is 3. As 7 is a factor of 28, the remainder when divided by 7 will be 3 SUFFICIENT

2 If n and m are positive integers, what is the remainder when 3^(4n + 2 + m) is divided by 10 ?(1) n = 2(2) m = 1

The Concept tested here is cycles of powers of 3.

The cycles of powers of 3 are : 3,9,7,1

St I) n = 2. This makes 3^(4*2 +2 + m) = 3^(10+m). we do not know m and hence cannot figure out the unit digit.

St II) m=1 . This makes 3^(4*n +2 + 1).4n can be 4,8,12,16... 3^(4*n +2 + 1) will be 3^7,3^11, 3^15,3^19 ..... in each case the unit digit will be 7. SUFFHence B

3.If p is a positive odd integer, what is the remainder when p is divided by 4 ?

(1) When p is divided by 8, the remainder is 5.

(2) p is the sum of the squares of two positive integers.

st1. take multiples of 8....divide them by 4...remainder =1 in each case...

st2. p is odd ,since p is square of 2 integers...one will be even and other odd....now when we divide any even square by 4 v ll gt 0 remainder..and when divide odd square vll get 1 as remainder......so intoatal remainder=1Ans : D

4.If p and n are positive integers and p > n, what is the remainder when p^2 - n^2 is divided by 15 ?




(1) The remainder when p + n is divided by 5 is 1.

(2) The remainder when p - n is divided by 3 is 1.

Ans: E

st1) p+n=6,11,16....insuff.st2) p-n=4,7,10....insuff...

multiply these two to get p^2-n^2.....multiplying any ttwo values from the above results in different remainder......

also can be done thru equation....p+n=5a+1..and so on

5.What is the remainder when the positive integer x is divided by 3 ?

(1) When x is divided by 6, the remainder is 2.

(2) When x is divided by 15, the remainder is 2.

Easy one , answer D

st 1...multiple of 6 will also be multiple of 3 so remainder wil be same as 2.

st2) multiple of 15 will also be multiple of 3....so the no.that gives remaindr 2 when divided by 15 also gives 2 as the remainder when divided by 3...

6.What is the remainder when the positive integer n is divided by 6 ?

(1) n is a multiple of 5.

(2) n is a multiple of 12.

Easy one. Answer B

st 1) multiples of 5=5,10,15....all gives differnt remainders with 6

st2)n is divided by 12...so it will be divided by 6...remainder=0




7If x, y, and z are positive integers, what is the remainder when 100x + 10y + z is divided by 7 ?

(1) y = 6

(2) z = 3

We need to know all the variables. We cannot get that from both the statements. Hence the answer is E.

8.If n is a positive integer and r is the remainder when 4 + 7n is divided by 3, what is the value of r ?

(1) n + 1 is divisible by 3.

(2) n > 20

Answer A st1... n+1 divisible by 3..so n=2,5,8,11......this gives 4+7n=18,39,60....remainder 0 in each case......st2) insufficient ....n can have any value

9.If n is a positive integer and r is the remainder when (n - 1)(n + 1) is divided by 24, what is the value of r ?

(1) n is not divisible by 2.

(2) n is not divisible by 3.

ST 1- if n is not divisible by 2, then n is odd, so both (n - 1) and (n + 1) are even. moreover, since every other even number is a multiple of 4, one of those two factors is a multiple of 4. so the product (n - 1)(n + 1) contains one multiple of 2 and one multiple of 4, so it contains at least 2 x 2 x 2 = three 2's in its prime factorization. But this is not sufficient, because it can be (n-1)*(n+1) can be 2*4 where remainder is 8. it can be 4*6 where the remainder is 0.

ST 2- if n is not divisible by 3, then exactly one of (n - 1) and (n + 1) is divisible by 3, because every third integer is divisible by 3. therefore, the product (n - 1)(n + 1) contains a 3 in its prime factorization. Just like st 1 this is not sufficient

the overall prime factorization of (n - 1)(n + 1) contains three 2's and a 3. therefore, it is a multiple of 24. sufficient




Answer C



bschooladmit.files.wordpress.com€¦ · Web view(2) The degree measure of angle ABC is twice the...

Documents

Transcript of bschooladmit.files.wordpress.com€¦ · Web view(2) The degree measure of angle ABC is twice the...