Freshmen Math Team Topics

Freshmen Math TopicsWritten by Kevin Hu, Naperville Central High School Class of 2011

Contents

1 Introduction 3

2 Number Theory and Divisibility 4

2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 LCM and GCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 LCM and GCD of multiple numbers . . . . . . . . . . . . . . . . . . . 5

2.2.2 The Euclidean algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Divisibility Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Trailing Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.6 Points of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Counting Basics and Simple Probability 10

3.1 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Miscellaneous Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12


4 Number Bases 14

4.1 b to 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2 10 to b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15


1

CONTENTS 2

5 Systems of Linear Equations and Inequalities 16

5.1 Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.2 The abs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.3 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17


Chapter 1

Introduction

The paper you are now reading should cover much of what you need to succeed in the2010-2011 math team season as a freshman. The chapters are based on topics from theNorth Suburban Math League contest, and will likely correspond to subjects in this sea-son’s ICTM (Illinois Council of Teachers of Mathematics) contests. The “Points of Inter-est” sections are not necessary for competition, but if you find a topic interesting, it maybe enriching to read the sections. The problems marked “(Open)” are open problems,meaning that they have not yet been solved by mathematicians.

If you have any questions or comments, please let me know.

Kevin Hu

[email protected]

This work was produced with LATEX 2ε, an open source typesetting system for scientificand mathematical documents of high typographic quality.

You are free to use any part of this work, but please attribute it appropriately.

3

Chapter 2

Number Theory and Divisibility

Number theory deals with properties of integers. It has been studied since the time ofancient Greece. Legendre and Gauss fathered modern number theory at the turn of thenineteenth century, and ever since, mathematicians have explored ideas like prime num-bers, Fermat’s Last Theorem, cryptography (codes), and programming. But before we getto advanced topics, let’s start basic.

2.1 Basics

We start with one of the most important facts in number theory.

Theorem 1. [Fundamental Theorem of Arithmetic]. Any integer greater than 1 can be writtenuniquely, up to the ordering of the factors, as the product of prime numbers.

Basically, this means that any integer greater than 1 has exactly one prime factorization.For example, the number 5 can be written as 51; the number 12 can be written as 22 × 3;the number 28 can be written as 22 × 7. This is basic, but crucial. Remember that everyinteger greater than 1 has a unique factorization. It cannot be written as the product ofprimes in any other way.

Now let’s introduce two terms: the least common multiple and the greatest commondivisor. The least common multiple of a and b, abbreviated as lcm(a, b) or sometimeswritten in shorthand as [a, b], is the smallest positive integer that is divisible by both aand b. You can calculate it as follows: Write the prime factorizations of a and b. Findevery distinct prime that divides either a or b. For each prime, look at what power of itdivides a and what power of it divides b. Take the larger of the two and write that powerdown. Repeat.

In mathematical form, we can say that if a = pe11 pe2

2 pe33 . . . and b = pf1

1 pf2

2 pf3

3 . . . where theexponents can be equal to zero and the pi are all of the prime numbers, then:

4

CHAPTER 2. NUMBER THEORY AND DIVISIBILITY 5

[a, b] = pmax(e1,f1)1 p

max(e2,f2)2 p

max(e3,f3)3 . . .

Similarly, the greatest common divisor or greatest common factor of a and b, abbreviatedas gcd(a, b) or sometimes written in shorthand as (a, b), is the largest positive integer thatdivides both a and b.

(a, b) = pmin(e1,f1)1 p

min(e2,f2)2 p

min(e3,f3)3 . . .

2.2 LCM and GCD

Recall the following: if a = pe11 pe2

2 pe33 . . . and b = pf1

1 pf2

2 pf3

3 . . . where the exponents can beequal to zero and the pi are all of the prime numbers, then:

[a, b] = pmax(e1,f1)1 p

max(e2,f2)2 p

max(e3,f3)3 . . .

(a, b) = pmin(e1,f1)1 p

min(e2,f2)2 p

min(e3,f3)3 . . .

Notice that max(ei, fi)+min(ei, fi) = ei +fi. Why? Because either ei < fi and max(ei, fi) =fi, min(ei, fi) = ei, or ei = fi and max(ei, fi) = min(ei, fi) = ei = fi, or ei > fi andmax(ei, fi) = ei, min(ei, fi) = fi. This leads to an interesting fact.

Theorem 2. [a, b]× (a, b) = a× b

2.2.1 LCM and GCD of multiple numbers

What about least common multiples and greatest common divisors of multiple numbers?The LCM and GCD have the following property:

Theorem 3. (a, b, c) = ((a, b), c) and [a, b, c] = [[a, b], c].

Now, notice that Theorem 2 can be written as [a, b] = a × b ÷ (a, b). This is suspiciouslysimilar to a formula you may know from set theory and Venn diagrams:

|A ∪ B| = |A|+ |B| − |A ∩ B|

whereA,B are sets, A∩B is their intersection, A∪B is their union, and |S| is the numberof elements in S.

This principle in set theory is known as the Principle of Inclusion and Exclusion, or thePIE. It can be extended to include more than just 2 sets. For three sets,


|X ∪ Y ∪ Z| = (|A|+ |B|+ |C|)− (|A ∩ B|+ |B ∩ C|+ |C ∩ A|) + |A ∩ B ∩ C|

Interestingly, we can say a similar fact for the LCM and GCD of three integers.

Theorem 4. [a, b, c] = a× b× c÷ ((a, b)× (b, c)× (c, a))× (a, b, c)

The relationship between set theory and LCM/GCD extends to any number of sets andthe corresponding number of integers.

2.2.2 The Euclidean algorithm

There is an alternate method for calculating a GCD by hand without prime factorization.This is useful if you’re trying to find the GCD of two very large numbers that don’t haveobvious factors.

Theorem 5. (a, b) = (a− kb, b)

where k is an integer.

This is known as the Euclidean algorithm. It makes the numbers you are trying to find theGCD of smaller; but, as it is an algorithm, it can require multiple steps. You are finishedwhen you find that one of the numbers is a factor of the other; then this number is theGCD. Let’s do a simple example. What is the GCD of 140 and 300?

(140, 300) = (140, 300− 2(140)) = (140, 300− 280) = (140, 20) = 20

As you can see, we stopped at (140, 20) because 20 is a factor of 140.

2.3 Divisibility Rules

The section title should not be interpreted as a complete sentence.

Divisibility rules are often taught as a list to memorize. While this is efficient, you shouldknow why the rules work.

Divisibility by 1: not worth explaining.

Divisibility by 2: the units digit is divisible by 2. This is because any positive integer canbe written as 10x + y where y is a positive integer and x is a nonnegative integer. 10x isdivisible by 2, so as long as y is divisible by 2, 10x + y is as well.

Divisibility by 3: the sum of the digits is divisible by 3. This is because any positiveinteger can be written as 10x + y where y is a positive integer and x is a nonnegative


integer. 10x + y = 9x + x + y, so as long as x + y is divisible by 3, so is 10x + y. If youcontinue this process, you find that if the sum of digits is divisible by 3, so is the number.

Divisibility by 4: the last two digits concatenated form a number divisible by 4. This isbecause 100 is divisible by 4.

Divisibility by 5: the last digit is 0 or 5.

Divisibility by 6: the number is divisible by 2 and by 3.

Divisibility by 7: the divisibility tests are a pain. If you really want one, write the digitsin reverse order, and multiply each digit in the new order by 1, 3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5,etc. Add these products together. If this is divisible by 7, so is the old number. (But hey, ifthe sum is still huge, you have to do it again. This is why dividing by 7 sucks.)

Divisibility by 8: the last three digits concatenated form a number divisible by 8. This isbecause 1000 is divisible by 8.

Divisibility by 9: the sum of digits is divisible by 9. This is because you can write thenumber in expanded notation, and every power of 10 is one more than a multiple of 9.

Divisibility by 10: the last digit is a 0.

Divisibility by 11: Remove the rightmost digit. Subtract it from the remaining number.Do this over and over again; if you get to a multiple of 11, the original is a multiple of 11.

The rest: Not worth memorizing.

2.4 Trailing Zeros

A math contest favorite: How many trailing zeros are there in x! Note that the “!” symbol isthe factorial (product of all positive integers less than or equal to x) sign, not the sign ofexcitement.

Well, for small numbers, you can do factorials out pretty easily. You can do 7! = 5040,8! = 40320, 9! = 362880, and 10! = 3628800, but then things get annoying, slow, andmistake-prone. So when a math contest asks you for the number of trailing zeros in 1000!,you better not try to multiply it out.

For each trailing zero, there must be a factor of 10. “So,” you think, “let’s just find howmany multiples of 10 there are from 1 to 1000.” But, alas, that would be inadequate,because 2 × 5 = 10, but 2 and 5 are not multiples of ten. Now what? Notice that we cancount the number of multiples of 5, because for each multiple of 5 there is inevitably amultiple of 2. Is that it?

No. Each multiple of 5 contributes 1 trailing zero, but each multiple of 25 contributes 2trailing zeros because 25× 4 = 100. Each multiple of 125 contributes 3 trailing zeros, andso on. Then, we find that the number of trailing zeros in x! is equal to:


⌊x

5

⌋+⌊ x

52

⌋+⌊ x

53

⌋+ . . .

2.5 Exercises

1. List all primes less than 100. How many are there?

2. Determine the prime factorizations of 2010 and 2011. Memorize these.

3. Prove Theorem 3. Hint: use prime factorizations.

4. Prove Theorem 4. Hint: use prime factorizations.

5. Evaluate [168, 144] by hand.

6. Evaluate (168, 144) by hand.

7. How do you use your calculator to find the GCD and LCM of two numbers?

8. How do you use your calculator to find the GCD and LCM of more than two num-bers?

9. Evaluate [720, 500]× (500, 720) by hand. Use prime factorizations.

10. Evaluate [720, 500]× (500, 720) by hand. Do not use prime factorizations.

11. Evaluate (144, 168, 180).

12. Evaluate [144, 168, 180].

13. Calculate (18084533, 20418113). The Euclidean Algorithm may be helpful.

14. How many trailing zeros are there in 1000!?

2.6 Points of Interest

1. How many right rectangular prisms are there with integral edge lengths, integralface diagonal lengths, and integral space diagonal lengths? In other words, howmany ordered triples (x, y, z) are there for which x, y, z,

√x2 + y2,

√y2 + z2,√

z2 + x2,√

x2 + y2 + z2 are all integers? (Open)

2. An infinite sum converges if it approaches a finite limit. Prove that the sum of thereciprocals of primes converges.

3. Twin primes are prime numbers that differ by two. Are there an infinite number ofpairs of twin primes? (Open)


4. Is every even number greater than 2 the sum of two primes? (Open)

5. Is there a prime between n and 2n inclusive for each positive n?

6. Is there a prime between n2 and (n + 1)2 for each positive n? (Open)

Chapter 3

Counting Basics and Simple Probability

The branch of mathematics that counting and probability fall under is called “Combina-torics.” It is the bane of contest mathematics students around the world. Students affec-tionately abbreviate it as “combo,” although it is usually accompanied with expletives ifresponsible adults are not around. What’s there to fear?

Combinatorics problems have a multitude of correct solution methods. Some are fasterthan others. The problem is that many combo problems have a multitude of incorrectsolution methods as well, which may be tantalizingly logical. As a math team competitor,your goal is to stay organized, stay logical, and stay alert.

3.1 Trees

You can use trees to organize outcomes in probability. For example, if a problem asks,What is the probability that you flip exactly 3 heads when you flip 6 coins?, if you don’t knowhow to proceed (ahem, see next section), you can use a tree. On every flip, you can geta head or a tail, so the tree branches out. Then along each “branch” of the tree, since thecoin has a 1/2 chance of getting a head or a tail, you multiply by 1/2 on each branch. Thisis only a very basic usage of trees.

In complex problems with multiple conditions, trees are more effective. Conditional prob-ability problems are an example. V. C. can press either the “g” or the “l” key on his computerkeyboard. His first press is random. If he presses the “l” key first, the probability he presses the“l” key again is 50%. If he presses the “g” key first, the probability he presses the “g” key againis 95%. V. C. presses two keys, one at a time. If the second press is a “g,” what’s the probabilityhe pressed “g” the first time? Here we draw a tree. There are two first-level branches —one for pressing “g” first, one for pressing “l” first. From each of these branches, there aretwo second-level branches — one for pressing “g,” and one for pressing “l.” We know hepresed “g” second, so we can get rid of the second-level branches with “l” presses. Theprobability he pressed “lg” is 50%× 50%, or 25%; we calculate this by multiplying across

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CHAPTER 3. COUNTING BASICS AND SIMPLE PROBABILITY 11

the branch. The probability he pressed “gg” is 50%× 95%, or 47.5%. Then the probabilityhe pressed “g” first is 47.5÷ (25 + 47.5) ≈ 65.5%.

There are better ways to do that problem, but trees are a good way to organize your work.Things get hairy when there are too many branches, or too many levels.

3.2 Permutations and Combinations

Let’s get the formulas over with first:

Theorem 6. nPk = P nk = n!

(n−k)!

Theorem 7. nCk = Cnk = C(n, k) =

(nk

)= n!

k!(n−k)!

A permutation is an arrangement of objects or values into a particular order. Thus, ordermatters. A combination is a grouping of objects or values. Thus, order does not matter.For example, the set {1, 2, 3} has permutations (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2),and (3, 2, 1). (Check that this is the correct number of permutations). On the other hand,the combinations of 2 numbers from that set are {1, 2}, {1, 3}, {2, 3}. Notice the differencein punctuation — a permutation has parentheses to denote that order matters (just likecoordinates have parentheses), while combinations have curly braces to denote that orderdoes not matter.

People often get confused about the meaning of “order matters” and “order does notmatter.” Clear up all this confusion: “order matters” means that the order is important; adifferent order just isn’t the same.

Make sure you use the correct formula. Always.

3.3 Miscellaneous Probability

To find the expected value, add together Event Value× Event Probability for each event.For example, suppose a lottery system is run so that 100 tickets are sold for 1 dollar each.There is 1 forty-dollar prize, 2 twenty-five-dollar prizes, and 3 three-dollar prizes. Whatis the expected value of profit if you enter the lottery? Well, in the event you do not win,you profit $ − 1; there is a 0.94 probability of this occurring. In the event that you win athree-dollar prize, you profit $2; there is a 0.03 probability of this occurring. In the eventthat you win a twenty-five-dollar prize, you profit $24; there is a 0.02 probability of thisoccurring. In the event that you win a forty-dollar prize, you profit $39; there is a 0.01probability of this occurring. Then your expected profit is:

(−1)(0.94) + (2)(0.03) + (24)(0.02) + (39)(0.01) = −.01

You expect to lose 1 penny for each ticket you buy.


3.4 Exercises

1. How many ways are there to arrange the letters “ABCDE”?

2. How many ways are there to misspell the word “MISSPELL” using all of the sameletters (e.g. MISPSELL, LLEPSMIS)? (Hint: Treat each S as a different letter, andtreat each L as a different letter. Then think about how much you’ve overcountedby.)

3. Did you make the error of counting “MISSPELL” as a misspelling? Watch out fortricks.

4. How many subsets are there in a set with 3 elements, including the empty set? In aset with 4 elements? n elements? Why?

5. Shuhao is playing basketball. His first shot has a 1/2 probability of scoring. If hemakes a shot, the probability he scores the next time is 3/4. If he misses a shot, theprobability he scores the next time is 1/10. What’s the expected value of his scoreafter 5 shots?

6. J. Z. is playing basketball. His first shot has a 1/10 probability of scoring. If hemakes a shot, the probability he scores the next time is 3/4. If he misses a shot, theprobability he scores the next time is 1/10. What’s the expected value of his scoreafter 5 shots?

7. In the game of Math Roulette, there are six index cards. Five of the cards have mathproblems on them. Two people play. One person starts by flipping over one of theindex cards. The next person flips another. They take turns until somebody gets thecard without a math problem on it. Of course, this person is the loser. Should youstart or should you go second to maximize your chances of winning?

8. What if there are 8 index cards with 7 problems? 10 cards with 9? 2n cards with2n− 1?


1. A casino has a 50.0028% chance of winning a hand of Blackjack (Dealer stands onsoft 17; player may double on any two cards, player may double after splitting,player may resplit aces, player gets late surrender).

2. Look up the rules of Craps and find the house edge.

3. Prove the Binomial Theorem without expanding: (x + y)n =(

n0

)xny0 +

(n1

)xn−1y1 +

· · ·+(

nn

)x0yn.

4. Stare at Pascal’s Triangle in awe.


5. Prove Pascal’s Identity using formulas:(

nk

)=(

n−1k−1

)+(

n−1k

). Then prove it concep-

tually.

6. Prove the Hockey Stick Identity:(

rr

)+(

r+1r

)+ · · ·+

(nr

)=(

n+1r+1

).

Chapter 4

Number Bases

We count in base 10 because we have ten fingers. What if we had a different number?What if we have to do math with strange things that only have 2 fingers? Oh wait, wealready do. They’re called computers.

In bases greater than 10, we use letters to represent digits. For example, the digits avail-able in base 16 (hexadecimal) are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. In mathcontest problems, make sure that your answer makes sense. Don’t answer “3” if a prob-lem asks you about what base a number is in when the number has the digit 9.

4.1 b to 10

Theorem 8. dndn−1 . . . d3d2d1d0 (the line above the d’s means that they should be read as digits,not factors) in base b is equal to d0 × b0 + d1 × b1 + d2 × b2 + · · ·+ dn × bn in base 10.

There, that wasn’t so difficult, was it?

4.2 10 to b

Theorem 9. dndn−1 . . . d3d2d1d0 in base 10 is equal to . . .

Right. It’s a lot more complicated now. To convert n from base 10 to base b, do thefollowing algorithm:

Divide n by the highest possible power of b that’s less than n. Write down the quotient(not including the remainder). Take the remainder, and divide that by the next highestpower of b. Write down the quotient (not including the remainder). Repeat on and on.

There, that was pretty terrible, wasn’t it?

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CHAPTER 4. NUMBER BASES 15

When you have to convert from base c to base d, where c, d 6= 10, unless you noticesomething special, convert from base c to base 10, and then from base 10 to base d.

Wait. Unless you notice something special. Like, for example, 16 is a power of 2, so couldthere be something special relating base 2 digits with base 16 digits?

4.3 Exercises

1. Convert 30 from base 8 to base 10.






7. Notice that 23a + 22b + 2c + d, where a, b, c, d are each either 0 or 1, is a binaryexpression. It’s also an integer between 0 and 15, inclusive. Moreover, each orderedquadruple (a, b, c, d) uniquely defines an integer between 0 and 15, inclusive. So, isthere something special relating base 2 digits with base 16 digits?

8. Convert 1.5 from base 10 to base 7.

9. How many bases b less than 6 are there for which 36 in base b is a perfect square?

10. Did you realize that in the above problem 36 is not a valid representation in a baseless than 6? Watch out for tricks.

11. Mr. Young is from Planet Q, which counts in base... oh, darn, I don’t remember!What I do know is that I gave Mr. Young 100 Qoins for a pencil that costs 11 Qoins,and he correctly gave me 10 Qoins in change. The reason I don’t remember whatbase Planet Q uses is that it might not be an integer... but of course, it’s still a validbase. Well, tell me, what base do they use?


1. The base is also known as the radix. Some mathematicians have explored havinga complex numbers as a radix. The radix 2i allows every number to be expressedwithout any positive/negative symbols.

’08 MOP The function f is a polynomial with coefficients in the set {0, 1, 2, 3}. Find, in termsof n the number of such polynomials f such that f(2) = n.

Chapter 5

Systems of Linear Equations andInequalities

In this chapter we get to the heart of algebra: systems. Normally, systems can be quiteeasy to solve, and some problems on contests will be a cinch. But there are a few wrenchesthrown in here and there.

5.1 Vocabulary

Definition 1. A consistent system of equations has at least one solution. It could have manysolutions, but it definitely has at least one.

Definition 2. An inconsistent system does not have any solutions at all.

Definition 3. A dependent system means that one of the equations gives no extra information. Iftwo lines are the same, then the system is dependent.

Definition 4. An independent system means that every equation is distinct, so that every line isdifferent.

5.2 The abs

The absolute value function should always be a warning that you should be alert. Itreturns the distance from a number to 0. The graph looks like a V, going through theorigin and symmetric about the y-axis.

It’s always a good idea to plug your answer into the problem to see if it makes sense.Check to make sure you don’t somehow have an absolute value of an expression equal toa negative number.

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CHAPTER 5. SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES 17

When dealing with absolute values, break everything down into cases. If you have |x|,break it down into x < 0 and x ≥ 0. If you have |x + 3|, break it down into x < −3 andx ≥ −3. If you have |x/y|, life gets tougher. Just break down the problem into cases whereexpressions inside absolute values are either less than zero or greater than/equal to zero.You might need a lot of cases — don’t be discouraged.

A few things to keep in mind:

Theorem 10. |x + y| ≤ |x|+ |y|

Theorem 11. |xy| = |x| × |y|

Theorem 10 might seem boring, but extend it to multiple dimensions, and you have theTriangle Inequality. If you treat x, y as vectors, imagine point A to be the origin and B tobe A + ~x and C to be A + ~y, then you get BC ≤ AB + AC.

5.3 Inequalities

Inequalities are also tricky. They’re essentially solved with the same methods as equa-tions, but beware of pesky sign changes. Multiplying or dividing by negative factorsalways flips the sign. Make sure you check your work on these. One way to quicklycheck your work is to plug in the boundary or boundaries from your solution set into theproblem. If you get the boundary of a condition described in the problem, that’s a goodsign. Then, plug in things inside your solution set into the problem; they should satisfythe conditions. Plug in things outside your solution set into the problem; they should notsatisfy the conditions. Choose easy numbers like 0.

5.4 Exercises

1. Draw a graph of a consistent system.

2. Draw a graph of an inconsistent system.

3. Draw a graph of a dependent system.

4. Draw a graph of an independent system.

5. Solve for x: |x| = −x. Try both graphing and casework.

6. How do you use the absolute value function on your calculator?

7. How do you graph inequalities on your calculator?

8. Solve for x: |x| = −x2.

CHAPTER 5. SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES 18

9. Solve for (x, y): 3x + 2y = 4 and 24x + 16y = 31.

10. The system 8a + b = 14 and −4b− 32a = M has a solution (a, b). Find M .


1. Look up Gaussian elimination with matrices. What is back substitution? How canyou tell if a multivariable system is dependent or independent?

2. How can you tell if a system has solutions?

3. What is a vector space? What is a vector space dimension?

4. What is a null space?

5. Learn collegiate Linear Algebra.

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