so the process eventually stops when the remainder becomes zero. If r1 = 0, then gcd(m, n) = n. Otherwise, rk = gcd(m, n), where rk is the last nonzero remainder and can be expressed as a linear combination of m and n by eliminating remainders.

Proof. Express rk as a linear combination of m and n by eliminating remainders in the equations from the second last equation up. Hence every common divisor of m and n divides rk . But rk is itself a common divisor of m and n (it divides every ri —work up through the equations). Hence rk = gcd(m, n).

gcd(12, 15) = 3. Note that 1 is relatively prime to every integer m. The following theorem collects three basic properties of relatively prime integers.

Theorem 11. If m and n are integers, not both zero:

(i) m and n are relatively prime if and only if 1 = xm + yn for some integers x and y.

dd

(iii)Suppose that m and n are relatively prime.

(a)If m|k and n|k, where k Z, then mn|k.

(b)If m|kn for some k Z, then m|k.

Proof. (i) If 1 = xm + yn with x, y Z, then every divisor of both m and n divides 1, so must be 1 or −1. It follows that gcd(m, n) = 1. The converse is by the euclidean algorithm.

(iii). Write 1 = xm + yn, where x, y Z. If k = am and k = bn, a, b Z, then k = kxm + kyn = (xb + ya)mn, and (a) follows. As to (b), suppose that kn = qm, q Z. Then k = kxm + kyn = (kx + qn)m, so m|k.

PRIME FACTORIZATION

Recall that an integer p is called a prime if:

(i)p 2.

(ii)The only positive divisors of p are 1 and p.

The reason for not regarding 1 as a prime is that we want the factorization of every integer into primes (as in Theorem 5) to be unique. The following result is needed.

Theorem 12. Euclid’s Lemma. Let p denote a prime.

(i)If p|mn where m, n Z, then either p|m or p|n.

(ii)If p|m1m2 · · · mr where each mi Z, then p|mi for some i.

Proof. (i) Write d = gcd(m, p). Then d|p, so as p is a prime, either d = p or d = 1. If d = p, then p|m; if d = 1, then since p|mn, we have p|n by

By Theorem 5, every integer n 2 can be written as a product of (one or more) primes. For example, 12 = 22 · 3, 15 = 3 · 5, 225 = 32 · 52. This factorization is unique.

Theorem 13. Prime Factorization Theorem. Every integer n 2 can be written as a product of (one or more) primes. Moreover, this factorization is unique except for the order of the factors. That is, if

n = p1p2 · · · pr and n = q1q2 · · · qs ,

where the pi and qj are primes, then r = s and the qj can be relabeled so that pi = qi for each i.

Proof. The existence of such a factorization was shown in Theorem 5. To prove uniqueness, we induct on the minimum of r and s. If this is 1, then n is a prime and the uniqueness follows from Euclid’s lemma. Otherwise, r 2 and

determined uniquely by n. If every ni = 1, we say that n is square-free, while if n has only one prime divisor, we call n a prime power.

If the prime factorization n = p1n1 p2n2 · · · prnr of an integer n is given, and if d is a positive divisor of n, then these pi are the only possible prime divisors of d (by Euclid’s lemma). It follows that

Corollary 14. If the prime factorization of n is n = p1n1 p2n2 · · · prnr , then the positive divisors d of n are given as follows:

d = p1d1 p2d2 · · · prdr where 0 di ni for each i.

This gives another characterization of the greatest common divisor of two positive integers m and n. In fact, let p1, p2, . . . , pr denote the distinct primes that divide one or the other of m and n. If we allow zero exponents, these numbers can be written in the form

It follows from Corollary 14 that the positive common divisors d of m and n

have the form

d = p1d1 p2d2 · · · prdr

where 0 di min(mi , ni ) for each i. [Here min(mi , ni ) denotes the smaller of the integers mi and ni .] Clearly then, we obtain gcd(m, n) if we set di = min(mi , ni ) for each i. Before recording this observation (in Theorem 15 below), we first consider a natural question: What if we use max(mi , ni ) for each exponent? [Here max(mi , ni ) is the larger of the integers mi and ni .] This leads to the dual of the notion of a greatest common divisor.

lcm(m, n), by

lcm(m, n) = p1max(m1 ,n1)p2max(m2,n2) · · · prmax(mr ,nr ).

It is clear by Corollary 14 that lcm(m, n) is a common multiple of m and n, and that it is a divisor of any such common multiple. Hence lcm(m, n) is indeed playing a role dual to that of the greatest common divisor. This discussion is summarized in

Theorem 15. Suppose that m and n are positive integers, and write

gcd(m, n) = p1min(m1,n1)p2min(m2,n2 ) · · · prmin(mr ,nr )

lcm(m, n) = p1max(m1 ,n1)p2max(m2,n2) · · · prmax(mr ,nr ).

The fact that max(m, n) + min(m, n) = m + n for any integers m and n gives immediately:

Corollary 16. mn = gcd(m, n)lcm(m, n) for all positive integers m and n.

Then gcd(600, 294) = 21315070 = 6, while lcm(600, 294) = 23315272 = 29,400. Note that Corollary 16 is verified by the fact that 600 · 294 = 6 · 29,400.

Of course, using Theorem 15 requires finding the prime factorizations of the integers m and n, and that is not easy. One useful observation is that if n 2 is

not a prime, then it has a prime factor p √n (it cannot have two factors greater

√

than n), so when looking for prime divisors of n it is only necessary to test the primes p √n. But for large integers, this is difficult, if not impossible. The euclidean algorithm (and Corollary 16) is a better method for finding greatest common divisors and least common multiples.

Note that this all generalizes: Given a finite collection a, b, c, . . . of positive integers, write them as

where the pi are the distinct primes that divide at least one of a, b, c, . . .. Then define their greatest common divisor and least common multiple as follows:

gcd(a, b, c, . . .) = p1min(a1,b1,c1 ,···)p2min(a2,b2,c2,···) · · · prmin(ar ,br ,cr ,···)

lcm(a, b, c, . . .) = p1max(a1,b1,c1,···)p2max(a2,b2,c2,···) · · · prmax(ar ,br ,cr ,···).

Then Theorem 15 extends as follows: gcd(a, b, c, . . .) is the common divisor of a, b, c, . . ., that is, a multiple of every such common divisor, and lcm(a, b, c, . . .) is the common multiple of a, b, c, . . ., that is, a divisor of every such common multiple.

This is as far as we go into number theory, the study of the integers, a subject that has fascinated mathematicians for centuries. There remain many unanswered questions, among them the celebrated Goldbach conjecture that every even number greater than 2 is the sum of two primes. This appears to be very difficult, but it is known that every sufficiently large even number is the sum of a prime and a number that is the product of at most two primes.

However, the twentieth century brought one resounding success. The fact that 32 + 42 = 52 shows that the equation ak + bk = ck has integer solutions if k = 2.

However, Fermat asserted that there are no positive integer solutions if k 3. He wrote a note in his copy of Arithmetica by Diophantus that “I have discovered a truly remarkable proof but the margin is to small to contain it.” The result became known as Fermat’s last theorem and remained open for 300 years. But in 1997, Andrew Wiles proved the result: He related Fermat’s conjecture to a problem in geometry, which he solved.