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**Extra info for Advances in Computers, Vol. 21**

**Example text**

Proof Suppose x − y = am then any common divisor of x and m is also a common divisor of y. From this the result is immediate. 2 The Ring of Integers Mod N Perhaps the easiest way to handle results on congruences is to place them in the framework of abstract algebra. To do this we construct, for each n > 0 a ring, called the ring of integers modulo n. We will follow this approach. However we note, that although this approach simplifies and clarifies many of the proofs, historically purely number theoretical proofs were given.

It follows that any reduced residue system modulo n has φ(n) elements. 3 A reduced residue system modulo 6 would be {1, 5}. We now develop a formula for φ(n). As is the theme of this book, we first determine a formula for prime powers and then paste back together via the fundamental theorem of arithmetic. 7 For any prime p and m > 0, φ( p m ) = p m − p m−1 = p m (1 − 1 ). p Proof Recall that if 1 ≤ a ≤ p then either a = p or (a, p) = 1. It follows that the positive integers less than p m which are not relatively prime to p m are precisely the multiples of p, that is, p, 2 p, 3 p, .

Proof Suppose (n − 1)! ≡ −1 mod n. If n were composite then n = mk with 1 < m < n−1 and 1 < k < n−1. If m = k then both m and k are included in (n−1)!. It follows that (n − 1)! is divisible by n so that (n − 1)! ≡ 0 mod n contradicting the assertion that (n − 1)! ≡ −1 mod n. If m = k = 2 then (n − 1)! ≡ 0 mod m which is not congruent to −1 mod m. Therefore, n must be prime. If m = k = 2 then n = 4 and (n − 1)! = 6 which is not congruent to −1 mod 4. 3 Units and the Euler Phi Function In a field F every nonzero element has a multiplicative inverse.