# SOLUTION: Use the binomial theorem and mathematical induction to show the following. Let p be a prime. Then for any integer a, we have a^p =a (mod p). The "=" sign should be congruence. I n

Algebra ->  Algebra  -> Divisibility and Prime Numbers -> SOLUTION: Use the binomial theorem and mathematical induction to show the following. Let p be a prime. Then for any integer a, we have a^p =a (mod p). The "=" sign should be congruence. I n      Log On

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 Click here to see ALL problems on Divisibility and Prime Numbers Question 6222: Use the binomial theorem and mathematical induction to show the following. Let p be a prime. Then for any integer a, we have a^p =a (mod p). The "=" sign should be congruence. I need help, I have no idea what they want.Answer by khwang(438)   (Show Source): You can put this solution on YOUR website! Proof by Math. Ind. a^p = a mod p for any (fixed prime p) (**) Basic:when a = 1, a^p = 1 mod p = a mod p. So, (**) is true for a = 1. Inductive Hypothesis, assume that when a = k,(**) is true. That is, k^p = k mod p Conside, (k+1)^p = E C(p,i) k^i (i =0,..,p) [E means summation,by the binomial Theorem] since p is a prime, for any i , 1 <= i <= p-1, p is a divisor of C(p,i), hence C(p,i) = 0 mod p for such i. So, we have C(p,i) k^i = 0 mod p, for all 1 <= i <= p-1 Hence, (k+1)^p = E C(p,i) k^i = k^p + 1 mod p = (k + 1) mod p. [ by the induction hypothesis] This means (**) is true for a = k+1 and the inductive proof is complete. This is an important fact about prime numbers . It is an direct result about the group Zp. Kenny