CHAPTER I I Greatest Common Divisors 1. Intro ducing greatest common divisors We call the integer a a common divisor of b and c, naturally enough, if it is a divisor that b and c have in common|that is, if a j b and a j c. Among all these common divisors, the largest one plays an imp ortant role in numb er theory, and so we give it a name. The largest numb er among all of the common divisors of b and c is called the greatest common divisor or gcd of b and c, and we denote it by gcdb; c Example. The divisors of 140 are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140 along with their negatives; the p ositive divisors of 168 are 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, and 168. The common divisors of 140 and 168 are the integers that app ear in b oth of these lists, namely 1, 2, 4, 7, 14, and 28. Therefore, their greatest common divisor gcd140; 168 equals 28. Example. The divisors of 15 are 1, 3, 5, and 15. The divisors of 0 are|well, all the integers! see Problem 1.2 Therefore, al l of the divisors of 15 are common divisors of 15 and 0, and so gcd15; 0 = 15. Before you try it and get into trouble, we'll tell you that gcd 0; 0 is unde ned|this is the only case where twointegers can have in nitely many common divisors. If b is a nonzero integer, then b only has nitely many divisors in fact, none of them are bigger than jbj, byFact 1.8, and the same for c;sowe see that if even one of b and c is nonzero, then it makes sense to talk ab out gcdb; c. Certainly 1 is always a common divisor of b and c as we showed in Problem 1.2, so at least we know that gcdb; c 1. Many textb o oks use the briefer notation b; c for the greatest common divisor of b and c. This saves space, although the p ossibility for confusion can arise, since p oints in the plane x; y are denoted with the same notation, for example. So we'll stick to the notation gcdb; c in these notes. We'll start this section right o with a theorem ab out greatest common divisors that we will use a lot in the future. Theorem 2.1 is probably the rst fact ab out numb ers you've come across in these notes that isn't simply intuitively true ::: we're seeing the b eginning of the go o d side of pro of-based mathematics, the side that tells us things ab out numb ers that we didn't already know. Theorem 2.1. Let b and c be integers, not both zero, and set d = gcd b; c. Then we can nd integers u and v such that d = bu + cv . 0 0 0 0 5 6 I I. GREATEST COMMON DIVISORS Proof: Consider the collection of all p ositiveintegers a that can b e written in the form bu + cv , for some integers u and v . This collection has at least one element: taking u = b and v = c, 2 2 we see that b + c is a p ositiveinteger that can b e written that way. Therefore, byFact 1.1, this collection has a smallest element s. Picktwointegers u and v such that s = bu + cv . 0 0 0 0 Let's try to show that s divides b. Using the Division Algorithm, we can write b = qs + r with 0 r<s.We're trying to show that s j b, and so byFact 1.10 all we need to show is that r =0.Well, supp ose not|supp ose that 0 <r<s.We can write r = b qs = b qbu + cv =b1 qu +cqv : 0 0 0 0 But this can't b e, b ecause then r would b e a p ositivenumber smal ler than s that can b e written in the form bu + cv ! Since it's imp ossible that 0 <r<s,we conclude that really r = 0, and so s divides b. The exact same argument shows that s divides c as well, and so s is a common divisor of b and c. Since d is the greatest common divisor of b and c, it's de nitely true that s d. On the other hand, we can show that d divides s! This is b ecause d j b and d j c, and so d divides every integer of the form bu + cv Fact 1.7|including s.Well, if d divides s then d jsj=sFact 1.8; and since s d and d s,we conclude that d = s. In particular, d = bu + cv , which is what wewanted to show. 0 0 We know that every common divisor of b and c is no bigger than gcdb; c ::: but in fact even more is true: Fact 2.2. Every common divisor of b and c divides gcdb; c. For instance, in the example on page 5, all of the common divisors 1, 2, 4, 7, 14, and 28 of 140 and 168 are not only less than or equal to gcd140; 168 = 28, but they all divide 28. And we can prove that this always happ ens. Proof: If we let d = gcdb; c, then from Theorem 2.1, there are integers u and v such that 0 0 d = bu + cv .Now let a be any common divisor of b and c. Since a j b,we can write b = as 0 0 for some integer s; and since a j c,we can similarly write c = at for some integer t. Hence d =asu +atv = asu + tv ; 0 0 0 0 which shows that a is a divisor of d. By now, wehave seen three di erentways of describing the same numb er gcdb; c: gcdb; c equals the largest of all the common divisors of b and c this was the de nition; gcdb; c equals the smallest numb er that can b e written in the form bu + cv for integers u and v this was Theorem 2.1; gcdb; c equals the p ositive common divisor of b and c, such that every common divisor of b and c divides it this was Fact 2.2. Actually,tobetechnically correct there is one detail ab out the last statement that we haven't proved yet. Can you see what it is? 1. INTRODUCING GREATEST COMMON DIVISORS 7 Example. From the equation 15 5 + 38 2 = 1, we see that 1 can b e written in the form 15u +38v. Since 1 is the smallest numb er of all, 1 is certainly the smallest numb er that can b e written in that form, and so we know that gcd15; 38 = 1. We can verify this by writing down all of the p ositive divisors of 15 they are 1, 3, 5, and 15 and 38 they are 1, 2, 19, and 38, and seeing that 1 is the only one in common. We've remarked that 1 and 1 are always common divisors of anytwointegers a and b.As it turns out, the situation where these are the only common divisors of b and c has great imp ortance in the study of numb er theory|de nitely imp ortant enough to warrant a name of its own: if a and b are twointegers such that gcda; b = 1, then wesay that a and b are relatively prime,orcoprime.For instance, the previous example shows that 15 and 38 are relatively prime. We can also say that 15 is relatively prime to 38 and vice versa. If we just restate what we've already said in this section using this new terminology,we obtain a consequence of Theorem 2.1 that will b e very useful to us: Consequence 2.3. Two integers a and b arerelatively prime if, and only if, thereare integers u and v such that au + bv =1. Here's an example of howwe can use this Consequence to prove something interesting: if n has no divisors other than 1 in common with a nor with b, then in fact n has no divisors other than 1 in common with the pro duct ab.Tosay this another way: Fact 2.4. The product of two integers coprime to a third integer is again coprime to the third integer: If gcda; n= 1 and gcdb; n= 1, then gcdab; n= 1. Proof: Because of Consequence 2.3, there are integers x and y such that ax + ny = 1, and 0 0 0 0 there are integers x and y such that bx + ny = 1. This means 1 1 1 1 1= 11=ax + ny bx + ny =abx x + nbx y + ax y + ny y : 0 0 1 1 0 1 1 0 0 1 0 1 How often do you get any mileage out of the fact that 1 1 = 1? Letting u = x x and 0 1 v = bx y + ax y + ny y ,we see that 1 can b e written in the form abu + nv . Therefore, 1 0 0 1 0 1 using Consequence 2.3 again, gcdab; n= 1. Fact 2.5. If a is relatively prime to b, and a j bc, then actual ly a j c.