Mathematics Questions

Guys I didn't forget your request, just was collecting good questions to post.

So here are some inequality and absolute value questions from my collection. Not every problem below is hard, but there are a few, which are quite tricky. Please provide your explanations along with the answers.

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

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-20.html#p653690

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


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


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


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


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


7. |x+2|=|y+2| what is the value of x+y?(1) xy2 y n


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


12. Is r=s?(1) -s1so x1both of which make the question no so sufficient

lagomez VP

Joined: 05 Mar 2008 Posts: 1484 Followers: 10 Kudos [?]: 166 [0], given: 31

Re: Inequality and absolute value questions from my collection[#permalink] 16 Nov 2009, 13:19 Bunuel wrote:

12. Is r=s?(1) -s1so x1both of which make the question no so sufficient

(1) (x-1)^2 0x cannot be -1 to 1 i.e. x1. NSF.

From 1 and 2: x is >1 but =s or r1

Statement 1:

2(1)-2(1/2)=1 , x,y are both positve

2(1/2)-2(-1/2)=1 x is positive, y is negative

INSUFFICIENT

Statement 2:

Either (x,y) are both positive or both negative

INSUFFICENT

Statement 1 and 2:

With both requirements x must be greater than y and satisfy this equation: 2x-2y=1

2(1)-2(1/2)=1 , x,y are both positve and x>y

2(1/2)-2(-1/2)=1 x is positive, y is negative and x>y

Answer: E

ichha148 Senior Manager

Joined: 16 Apr 2009 Posts: 348 Followers: 1 Kudos [?]: 25 [0], given: 10

Re: Inequality and absolute value questions from my collection[#permalink] 17 Nov 2009, 10:27 12. Is r=s?

(1) -s= 0.86 i.e. 1 >= 0.86Combining , any values can be taken , on values > =1 , both r and swill be same

3. Is x^2 + y^2 > 4a?

(1) (x + y)^2 = 9a

(2) (x y)^2 = aC is the answer

Combined both and the equation will give x^2 + y^2 = 5a _________________Always tag your question

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h2polo Manager

Joined: 13 Aug 2009 Posts: 207 Schools: Sloan '14 (S) Followers: 3 Kudos [?]: 60 [1] , given: 16

Re: Inequality and absolute value questions from my collection[#permalink] 17 Nov 2009, 10:34 1 This post receivedKUDOSBunuel wrote:5. What is the value of y?(1) 3|x^2 -4| = y - 2 (2) |3 - y| = 11

Statement 1:

Two equations, two unknowns... INSUFFICIENT

Statement 2:

|3 - y| = 11(3-y)=11 or (3-y)=-11y=-8, 14

INSUFFICIENT

Statements 1 and 2:

y must be 14 because 3|x^2 -4| can never be a negative value (no matter what you plug in for x, you will get a positve value because of the absolute value signs).

SUFFICIENT

ANSWER: C.

Last edited by h2polo on 17 Nov 2009, 10:54, edited 1 time in total.

h2polo Manager

Joined: 13 Aug 2009 Posts: 207 Schools: Sloan '14 (S) Followers: 3 Kudos [?]: 60 [0], given: 16

Re: Inequality and absolute value questions from my collection[#permalink] 17 Nov 2009, 10:43 Bunuel wrote:6. If x and y are integer, is y > 0? (1) x +1 > 0 (2) xy > 0

Statement 1:

Nothing about y... INSUFFICIENT

Statement 2:

two equations, two unknowns... INSUFFICIENT

Statements 1 and 2:

From x +1 > 0 and the fact that x must be an integer, we know that x must be [0,1,2,3...]

Because we know that xy > 0, we know that x cannot be 0... therefore y must be a positive integer!

SUFFICIENT

ANSWER: C.

Marco83 Intern

Joined: 08 Nov 2009 Posts: 48 Followers: 0 Kudos [?]: 10 [0], given: 0 Re: Inequality and absolute value questions from my collection[#permalink] 17 Nov 2009, 10:48 4)I) 2x-2y=1 so y=x-1/2 NSII)x/y>0 so x and y have the same sign and the modulus of x has to be larger than the modulus of y NSTogether, to satisfy both clues needs to be larger than 1/2 and x becomes larger than 0; the stem is true, therefore C

Marco83 Intern

Joined: 08 Nov 2009 Posts: 48 Followers: 0 Kudos [?]: 10 [0], given: 0 Re: Inequality and absolute value questions from my collection[#permalink] 17 Nov 2009, 10:53 h2polo wrote:4. Are x and y both positive?(1) 2x-2y=1(2) x/y>1

Statement 1:

2(1)-2(1/2)=1 , x,y are both positve

2(1/2)-2(-1/2)=1 x is positive, y is negative

INSUFFICIENT

Statement 2:

Either (x,y) are both positive or both negative

INSUFFICENT

Statement 1 and 2:

With both requirements x must be greater than y and satisfy this equation: 2x-2y=1

2(1)-2(1/2)=1 , x,y are both positve and x>y

2(1/2)-2(-1/2)=1 x is positive, y is negative and x>y

Answer: E

Your last choice of numbers: x=1/2, y=-1/2 does not satisfy clue I, because 2*(1/2)-2*(-1/2)=2, not 1

Please find below new set of PS problems:

1. A family consisting of one mother, one father, two daughters and a son is taking a road trip in a sedan. The sedan has two front seats and three back seats. If one of the parents must drive and the two daughters refuse to sit next to each other, how many possible seating arrangements are there? (A) 28 (B) 32 (C) 48 (D) 60 (E) 120

2. What is the probability that a 3-digit positive integer picked at random will have one or more "7" in its digits?(A) 271/900(B) 27/100 (C) 7/25 (D) 1/9 (E) 1/10

3. A sphere is inscribed in a cube with an edge of 10. What is the shortest possible distance from one of the vertices of the cube to the surface of the sphere? (A 10( sqrt3- 1) (B) 5 (C) 10( sqrt2 - 1) (D) 5( sqrt3 - 1) (E) 5( sqrt2 - 1)

4. A contractor estimated that his 10-man crew could complete the construction in 110 days if there was no rain. (Assume the crew does not work on any rainy day and rain is the only factor that can deter the crew from working). However, on the 61-st day, after 5 days of rain, he hired 6 more people and finished the project early. If the job was done in 100 days, how many days after day 60 had rain? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8

5. If s and t are positive integer such that s/t=64.12, which of the following could be the remainder when s is divided by t? (A) 2 (B) 4 (C) 8 (D) 20 (E) 45

6. A committee of 6 is chosen from 8 men and 5 women so as to contain at least 2 men and 3 women. How many different committees could be formed if two of the men refuse to serve together?(A) 3510(B) 2620(C) 1404(D) 700(E) 635

7. If x is positive, which of the following could be the correct ordering of 1/x,2x and x^2 ? I. x^2 6412 div 100 Remainder = 12= 3206/50 => 3206 div 50 Remainder = 6= 1603/25 => 1603 div 25 Remainder = 3No more common factors.I don't see how the remainder could be anything but 3,6,12,24,48. What am I doing wrong here?

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yangsta8 Senior Manager

Joined: 31 Aug 2009 Posts: 425 Location: Sydney, Australia Followers: 6 Kudos [?]: 78 [2] , given: 20

Re: Good set of PS 2[#permalink] 16 Oct 2009, 21:40 2 This post receivedKUDOSBunuel wrote:6. A committee of 6 is chosen from 8 men and 5 women so as to contain at least 2 men and 3 women. How many different committees could be formed if two of the men refuse to serve together?(A) 3510(B) 2620(C) 1404(D) 700(E) 635

The committee can be formed in two ways:1) 2 men and 4 women2) 3 men and 3 womenThe answer is the sum of these.

1) 2 men and 4 women = (8C2 - 1) x 5C4 = 27 x 5 = 135Subtract 1 since there is one combo of men that are not allowed.2) 3 men and 3 women = (8C3 - 6) x 5C3 = (56-6) x 10 = 500Subtract 6 since there are 6 groups of men that can include those the two that refuse to work together.Adding these together we get 135+500 = 635ANS = E

Bunuel GMAT Club team member

Joined: 02 Sep 2009 Posts: 12187 Followers: 1883 Kudos [?]: 10238 [0], given: 989

Re: Good set of PS 2[#permalink] 16 Oct 2009, 21:45 yangsta8 wrote:Bunuel wrote:5. If s and t are positive integer such that s/t=64.12, which of the following could be the remainder when s is divided by t? (A) 2 (B) 4 (C) 8 (D) 20 (E) 45

s/t = 64.12 = 6412/100 => 6412 div 100 Remainder = 12= 3206/50 => 3206 div 50 Remainder = 6= 1603/25 => 1603 div 25 Remainder = 3No more common factors.I don't see how the remainder could be anything but 3,6,12,24,48. What am I doing wrong here?

This is a good question. Well, you did everything right, though it could be done easier, but conclusion is not correct. Look again at the remainders... You should see the pattern. _________________NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

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yangsta8 Senior Manager

Joined: 31 Aug 2009 Posts: 425 Location: Sydney, Australia Followers: 6 Kudos [?]: 78 [1] , given: 20

Re: Good set of PS 2[#permalink] 16 Oct 2009, 21:46 1 This post receivedKUDOSBunuel wrote:7. If x is positive, which of the following could be the correct ordering of 1/x,2x and x^2 ? I. x^2 [i];Amt saved this year * (1+r) = amount avail to spend next year, so Sa(1+r) = Sp2. Given Sp2 = Sp1/2, Sp1/2 = Sa(1+r) or Sp1 = 2*Sa*(1+r) -> (ii)Combining (i) and (ii), I = Sa + 2*Sa*(1+r) or I = Sa*(1+2+2r) so Sa/I = 1/(3+2r).

7. Before being simplified, the instructions for computing income tax in Country Rwere to add 2 percent of one's annual income to the average(arithmetic mean)of 100units of Country R's currency and 1 percent of one's annual income. Which of the following represents the simplified formula for computing the income tax in Country R's currency, for a person in that country whose annual income is I? (C) 50+I/40: Not sure I understand this correctly, but I'll give it a try anyway. T = 0.02*I + (100+0.01*I)/2 = 0.025*I+50 = I/40 + 50.

8. How many positive integers less than 10,000 are such that the product of their digits is 210?(B) 30 : Boy this is tough; 210 = 2*3*5*7. If we take all 4 primes as separate digits, then 4*3*2*1 = 24 different #'s. We can also make #'s from the digits 6 (2*3), 5 and 7 = 3*2*1 = 6 different #'s so total 30 #'s. Any other combination of these primes will give a digit > 9 and hence will not get the required result.

9. Find the number of selections that can be made taking 4 letters from the word"ENTRANCE". (A) 70 (B) 36 (C) 35 (D) 72 (E) 32: Not getting the answer .. I thought it should be 7*6*5*4 since 7 letters and 4 spots.

Find in the above word, the number of arrangements using the 4 letters.

10. How many triangles with positive area can be drawn on the coordinate plane such that the vertices have integer coordinates (x,y) satisfying 1x3 and 1y3? (B) 76: 9 possible options for vertices, need to chose any three to make a triangle so 9C3 = 84. However, 8 (3 along the length, 3 along the height and 2 diagonals)of these 3 sets of points will not make a triangle since they are in a straight line so 84-8 = 76.

rohitbhotica Intern

Joined: 07 Oct 2009 Posts: 19 Followers: 0 Kudos [?]: 3 [0], given: 0 Re: NEW SET of good PS(3)[#permalink] 21 Oct 2009, 10:31 badgerboy wrote:Ill take a shot -

1. ABCDE is a regular pentagon with F at its center. How many different triangles can be formed by joining 3 of the points A,B,C,D,E and F?(C) 20: 6 vertices, 3 to chose from so 6C3 = 20.

2. The function f is defined for all positive integers n by the following rule: f(n) is the number of positive integers each of which is less than n and has no positive factor in common with n other than 1. If p is prime, then f(p) = (B) P-2 : for a prime p, all of the numbers preceding it (except 1 will not be a factor of p). Since there p-1 #'s preceding it and we dont count 1, f(p) = p-1-1 = p-2.

3. How many numbers that are not divisible by 6 divide evenly into 264,600? (D) 63: used Bunuels trick. I'll let him explain since he was the one who helped me with this.

4.A certain quantity is measured on two different scales, the R-scale and the S-scale, that are related linearly. Measurements on the R-scale of 6 and 24 correspond to measurements on the S-scale of 30 and 60, respectively. What measurement on the R-scale corresponds to a measurement of 100 on the S-scale? (C) 48 : Let R = mS + c. Then 6 = m*30 + c and 24 = m*60+c; substituting for c, c = 6-30*m we get 24 = 60*m + 6-30*m, so m = 18/30 = 3/5. Solving for c, c = -12. So for S = 100, R = 3/5*100 -12 = 48.

5. Mrs. Smith has been given film vouchers. Each voucher allows the holder to see a film without charge. She decides to distribute them among her four nephews so that each nephew gets at least two vouchers. How many vouchers has Mrs. Smith been given if there are 120 ways that she could distribute the vouchers?(A) 13(B) 14(C) 15(D) 16(E) more than 16: No idea.

6. This year Henry will save a certain amount of his income, and he will spend the rest. Next year Henry will have no income, but for each dollar that he saves this year, he will have 1 + r dollars available to spend. In terms of r, what fraction of his income should Henry save this year so that next year the amount he was available to spend will be equal to half the amount that he spends this year? (E) 1/(2r+3): Let I = income earned, Sa = amt saved, Sp1 = amt avail. to spend this year and Sp2 = amt avail. to spend next year.Need to find Sa/I such that Sp2 = Sp1/2.I = Sa + Sp1 -> [i];Amt saved this year * (1+r) = amount avail to spend next year, so Sa(1+r) = Sp2. Given Sp2 = Sp1/2, Sp1/2 = Sa(1+r) or Sp1 = 2*Sa*(1+r) -> (ii)Combining (i) and (ii), I = Sa + 2*Sa*(1+r) or I = Sa*(1+2+2r) so Sa/I = 1/(3+2r).

7. Before being simplified, the instructions for computing income tax in Country Rwere to add 2 percent of one's annual income to the average(arithmetic mean)of 100units of Country R's currency and 1 percent of one's annual income. Which of the following represents the simplified formula for computing the income tax in Country R's currency, for a person in that country whose annual income is I? (C) 50+I/40: Not sure I understand this correctly, but I'll give it a try anyway. T = 0.02*I + (100+0.01*I)/2 = 0.025*I+50 = I/40 + 50.

8. How many positive integers less than 10,000 are such that the product of their digits is 210?(B) 30 : Boy this is tough; 210 = 2*3*5*7. If we take all 4 primes as separate digits, then 4*3*2*1 = 24 different #'s. We can also make #'s from the digits 6 (2*3), 5 and 7 = 3*2*1 = 6 different #'s so total 30 #'s. Any other combination of these primes will give a digit > 9 and hence will not get the required result.

9. Find the number of selections that can be made taking 4 letters from the word"ENTRANCE". (A) 70 (B) 36 (C) 35 (D) 72 (E) 32: Not getting the answer .. I thought it should be 7*6*5*4 since 7 letters and 4 spots.


10. How many triangles with positive area can be drawn on the coordinate plane such that the vertices have integer coordinates (x,y) satisfying 1x3 and 1y3? (B) 76: 9 possible options for vertices, need to chose any three to make a triangle so 9C3 = 84. However, 8 (3 along the length, 3 along the height and 2 diagonals)of these 3 sets of points will not make a triangle since they are in a straight line so 84-8 = 76.

I think your answer for the 8th question is wrong coz ur missing the cases when it is 5, 6 ,7 and 1 which are 24 more cases. So the answer should be 54 and not 30

For question 5 assume first that Mrs. Smith has given 8 tickets to her grandsons by giving 2 to each and has "x" tickets left. So she can now distribute these x tickets to her 4 grandsons in (x+3)C3 ways. This is selection without arrangement so we use this formula.We thus get this to be equal to 120. Thus we get x+3 = 10 and x = 7hence total tickets is 15

For the 9th question we have to take 3 cases1) where only one of each letter is chosen = 6C3 ways = 15 ways2) Where either 2 E's or 2N's are chosen = 2 * 5C2 = 20 ways3) Where 2 E's and 2 N's are chosen = 1 way

Hence answer = 36 ways

for the second question I think we should count 1, as there is no reason for not counting it and hence the answer should be p-1 only.

badgerboy Manager

Joined: 08 Oct 2009 Posts: 68 Followers: 1 Kudos [?]: 11 [0], given: 5 Re: NEW SET of good PS(3)[#permalink] 21 Oct 2009, 11:17 rohitbhotica wrote:I think your answer for the 8th question is wrong coz ur missing the cases when it is 5, 6 ,7 and 1 which are 24 more cases. So the answer should be 54 and not 30





Shoot ... good catch on Q8. I forgot about the 1.Can you explain why the (x+3)C3 for Q5?Thanks for the explanations .. they were very helpful.

rohitbhotica Intern

Joined: 07 Oct 2009 Posts: 19 Followers: 0 Kudos [?]: 3 [0], given: 0 Re: NEW SET of good PS(3)[#permalink] 21 Oct 2009, 11:21 badgerboy wrote:rohitbhotica wrote:I think your answer for the 8th question is wrong coz ur missing the cases when it is 5, 6 ,7 and 1 which are 24 more cases. So the answer should be 54 and not 30





Shoot ... good catch on Q8. I forgot about the 1.Can you explain why the (x+3)C3 for Q5?Thanks for the explanations .. they were very helpful.

no i think 1 has to be counted. What is the OA?

badgerboy Manager

Joined: 08 Oct 2009 Posts: 68 Followers: 1 Kudos [?]: 11 [0], given: 5 Re: NEW SET of good PS(3)[#permalink] 21 Oct 2009, 11:26 rohit,agreed 1 has to be counted. i read the question a second time and it made sense. I shall defer to Bunuel for OA's.


Joined: 02 Sep 2009 Posts: 12187 Followers: 1883 Kudos [?]: 10238 [0], given: 989

Re: NEW SET of good PS(3)[#permalink] 23 Oct 2009, 18:40 ANSWERS (OAs):

As most of the problems was solved correctly, I'm posting only OAs. Please let me know if anyone needs any clarification.

1. ABCDE is a regular pentagon with F at its center. How many different triangles can be formed by joining 3 of the points A,B,C,D,E and F?(A) 10(B) 15(C) 20(D) 25(E) 30

Answer: C.

2. The function f is defined for all positive integers n by the following rule: f(n) is the number of positive integers each of which is less than n and has no positive factor in common with n other than 1. If p is prime, then f(p) = (A) P-1 (B) P-2 (C) (P+1)/2 (D) (P-1)/2 (E) 2

Answer: A.

3. How many numbers that are not divisible by 6 divide evenly into 264,600? (A) 9 (B) 36 (C) 51 (D) 63 (E) 72

Answer: D.

4.A certain quantity is measured on two different scales, the R-scale and the S-scale, that are related linearly. Measurements on the R-scale of 6 and 24 correspond to measurements on the S-scale of 30 and 60, respectively. What measurement on the R-scale corresponds to a measurement of 100 on the S-scale? (A) 20 (B) 36 (C) 48 (D) 60 (E) 84

Answer: C.

5. Mrs. Smith has been given film vouchers. Each voucher allows the holder to see a film without charge. She decides to distribute them among her four nephews so that each nephew gets at least two vouchers. How many vouchers has Mrs. Smith been given if there are 120 ways that she could distribute the vouchers?(A) 13(B) 14(C) 15(D) 16(E) more than 16

Answer: C.

6. This year Henry will save a certain amount of his income, and he will spend the rest. Next year Henry will have no income, but for each dollar that he saves this year, he will have 1 + r dollars available to spend. In terms of r, what fraction of his income should Henry save this year so that next year the amount he was available to spend will be equal to half the amount that he spends this year? (A) 1/(r+2)(B) 1/(2r+2)(C) 1/(3r+2) (D) 1/(r+3) (E) 1/(2r+3)

Answer: E.

7. Before being simplified, the instructions for computing income tax in Country Rwere to add 2 percent of one's annual income to the average(arithmetic mean)of 100units of Country R's currency and 1 percent of one's annual income. Which of the following represents the simplified formula for computing the income tax in Country R's currency, for a person in that country whose annual income is I? (A) 50+I/200(B) 50+3I/100(C) 50+I/40 (D) 100+I/50 (E) 100+3I/100

Answer: C.

8. How many positive integers less than 10,000 are such that the product of their digits is 210?(A) 24 (B) 30 (C) 48 (D) 54 (E) 72

Answer: D.

9. Find the number of selections that can be made taking 4 letters from the word"ENTRANCE". (A) 70 (B) 36 (C) 35 (D) 72 (E) 32

Answer:B.


Answer:606.

10. How many triangles with positive area can be drawn on the coordinate plane such that the vertices have integer coordinates (x,y) satisfying 1x3 and 1y3? (A) 72 (B) 76 (C) 78 (D) 80 (E) 84

Answer: B. _________________NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!







Last edited by Bunuel on 02 Nov 2009, 15:12, edited 1 time in total.

Economist Director

Joined: 01 Apr 2008 Posts: 918 Schools: IIM Lucknow (IPMX) - Class of 2014 Followers: 8 Kudos [?]: 125 [0], given: 18

Re: NEW SET of good PS(3)[#permalink] 23 Oct 2009, 22:39 Hi Bunuel, would appreciate if you can explain the solutions for 3,5 and 9.

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Joined: 02 Sep 2009 Posts: 12187 Followers: 1883 Kudos [?]: 10238 [7] , given: 989

Re: NEW SET of good PS(3)[#permalink] 23 Oct 2009, 23:00 7 This post receivedKUDOSEconomist wrote:Hi Bunuel, would appreciate if you can explain the solutions for 3,5 and 9.

3. How many numbers that are not divisible by 6 divide evenly into 264,600? (A) 9 (B) 36 (C) 51 (D) 63 (E) 72

Answer: D.

First of all you should know the formula counting the number of distinc factors of an integer:

You have to write the number as the product of primes as a^p*b^q*c^r, where a, b, and c are prime factors and p,q, and r are their powers.

The number of factors the number contains will be expressed by the formula (p+1)(q+1)(r+1).Let's take an example for clear understanding:Find the number of all (distinct) factors of 1435:1. 1435 can be expressed as 5^1*17^1*19^1 2. total number of factors of 1435 including 1 and 1435 itself is (1+1)*(1+1)*(1+1)=2*2*2=8 factors.

ORDistinct factors of 18=2*3^2 --> (1+1)*(2+1)=6. Lets check: factors of 18 are: 1, 2, 3, 6, 9 an 18 itself. Total 6.

Back to our question:How many numbers that are not divisible by 6 divide evenly into 264,600?

264,600=2^3*3^3*5^2*7^2

We should find the factor which contain no 2 and 3 together, so not to be divisible by 6.

Clearly, the factors which contain only 2,5,7 and 3,5,7 won't be divisible by 6. So how many such factors are there?2^3*5^2*7^2 --> (3+1)*(2+1)*(2+1)=36 (the product of powers of 2, 5,and 7 added 1)

3^3*5^2*7^2 --> (3+1)*(2+1)*(2+1)=36 (the product of powers of 2, 5, and 7 added 1)

So 36+36=72. BUT this number contains duplicates:

For example: 2^3*5^2*7^2--> (3+1)*(2+1)*(2+1)=36 This 36 contains the factors when the power of 2 is 0 (2^0=1)--> 2^0*5^2*7^2 giving us only the factors which contain 5-s and/or 7-s. (5*7=35, 5*7^2=245, 5^2*7=175, 5*7^0=5, 5^0*7=7....) number of such factors are (2+1)*(2+1)=9 (the product of powers of 5 and 7 added 1).

And the same factors are counted in formula 3^3*5^2*7^2 --> (3+1)*(2+1)*(2+1)=36: when power of 3 is 0 (3^0=1). --> 5*7=35, 5*7^2=245, 5^2*7=175, 5*7^0=5, 5^0*7=7.... such factors are (2+1)*(2+1)=9. (the product of powers of 5 and 7 added 1).

So we should subtract this 9 duplicated factors from 72 --> 72-9=63. Is the correct answer.

The problem can be solved from another side:264,600=2^3*3^3*5^2*7^2 # of factors= (3+1)(3+1)(2+1)(2+1)=144. So our number contains 144 distinct factors. # of factors which contain 2 and 3 is 3*3=9 (2*3, 2^2*3, 2^3*3, 2*3^2, 2^2*3^2, 2^3*3^2, 2*3^3, 2^2*3^3, 2^3*3^3 total 9) multiplied by (2+1)*(2+1)=9 (powers of 5 and 7 plus 1) --> 9*9=81 ---> 144-81=63.

Hope now it's clear. _________________NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!







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Re: NEW SET of good PS(3)[#permalink] 23 Oct 2009, 23:42 5 This post receivedKUDOS5. Mrs. Smith has been given film vouchers. Each voucher allows the holder to see a film without charge. She decides to distribute them among her four nephews so that each nephew gets at least two vouchers. How many vouchers has Mrs. Smith been given if there are 120 ways that she could distribute the vouchers?(A) 13(B) 14(C) 15(D) 16(E) more than 16

Answer: C.

Clearly there are more than 8 vouchers as each of four can get at least 2. So, basically 120 ways vouchers can the distributed are the ways to distribute vouchers, so that each can get from zero to as at "least 2", or 2*4=8, we already booked. Let be .

In how many ways we can distribute identical things among 4 persons? Well there is a formula for this but it's better to understand the concept.

Let . And imagine we want to distribute 5 vouchers among 4 persons and each can get from zero to 5, (no restrictions).

Consider:

We have 5 tickets (t) and 3 separators between them, to indicate who will get the tickets:

Means that first nephew will get all the tickets,

Means that first got 0, second 1, third 3, and fourth 1

And so on.

How many permutations (arrangements) of these symbols are possible? Total of 8 symbols (5+3=8), out of which 5 's and 3 's are identical, so . Basically it's the number of ways we can pick 3 separators out of 5+3=8: .

So, # of ways to distribute 5 tickets among 4 people is .

For it will be the same: # of ways to distribute tickets among 4 persons (so that each can get from zero to ) would be .

. --> . Plus the 8 tickets we booked earlier: .

Answer: C (15).

P.S. Direct formula:

The total number of ways of dividing n identical items among r persons, each one of whom, can receive 0,1,2 or more items is .

The total number of ways of dividing n identical items among r persons, each one of whom receives at least one item is .

Hope it helps. _________________NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!







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Joined: 08 Oct 2009 Posts: 72 Location: Denmark, Europe Schools: Darden Class of 2012 Followers: 1 Kudos [?]: 10 [0], given: 6 Re: NEW SET of good PS(3)[#permalink] 24 Oct 2009, 13:06 yangsta8 wrote:Bunuel wrote:1. ABCDE is a regular pentagon with F at its center. How many different triangles can be formed by joining 3 of the points A,B,C,D,E and F?(A) 10(B) 15(C) 20(D) 25(E) 30

6 points in total to make triangles. I think a combination of any 3 will make a unique triangle so:6C3 = 20

8.Forensic scientists use the equationh = 2.6f + 47.2to estimate the height, h, of a woman given the length in centimeters, f, of her femur bone. Suppose the equation has a margin of error of + 4 centimeters and the length of a female skeleton's femur is 48 centimeters. Write and solve an absolute value inequality that describes the woman's height in centimeters.

Evaluate:1) log497 log8642) Solve for x:3) 2 logbx = 2 logb(1 a) +2 logb(1 + a) logb4) logbx = 2 a + logbAnswers:1) 2) 3) a 4) ahttp://edhelper.com/logarithms.htm

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