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230hw6sol - Math 230 E Homework 6 Fall 2011 Drew Armstrong...

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Math 230 E Fall 2011 Homework 6 Drew Armstrong Problem 1. Without using Fermat’s little Theorem , prove the Freshman’s Binomial Theorem : For all a, b, p Z with p prime we have ( a + b ) p a p + b p mod p. Proof. By the Binomial Theorem we have ( a + b ) p = a p + p 1 a p - 1 b + p 2 a p - 2 b 2 + · · · + p p - 2 a 2 b p - 2 + p p - 1 ab p - 1 + b p , so we will be done if we can show that ( p k ) 0 mod p for all 0 < k < p ; in other words, that p divides the integer ( p k ) . Well, we know that p divides p !, hence it must divide the product p ! = ( p k ) ( k !( p - k )!). However, if 0 < k < p , then k !( p - k )! is a product of integers all of which are strictly smaller than p . If p divides k !( p - k )!, then by Euclid’s Lemma, p divides some number strictly smaller than itself, which is a contradiction to Lemma 2.11(iv) on page 25 of the text. Hence p does not divide k !( p - k )!. But then since p divides the product ( p k ) ( k !( p - k )!), Euclid’s Lemma implies that p divides ( p k ) . Problem 2. Recreate Euler’s (1736) proof of Fermat’s little Theorem. That is, fix a prime p Z and use induction on n to show that n p n mod
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