document.write( "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. \n" ); document.write( "
Algebra.Com's Answer #3355 by khwang(438)\"\" \"About 
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Proof by Math. Ind. a^p = a mod p for any (fixed prime p) (**)
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\n" ); document.write( " Basic:when a = 1, a^p = 1 mod p = a mod p.
\n" ); document.write( " So, (**) is true for a = 1.
\n" ); document.write( " Inductive Hypothesis, assume that when a = k,(**) is true.
\n" ); document.write( " That is, k^p = k mod p
\n" ); document.write( " Conside, (k+1)^p = E C(p,i) k^i (i =0,..,p)
\n" ); document.write( " [E means summation,by the binomial Theorem]
\n" ); document.write( " since p is a prime, for any i , 1 <= i <= p-1, p is a divisor of C(p,i),
\n" ); document.write( " hence C(p,i) = 0 mod p for such i.
\n" ); document.write( " So, we have C(p,i) k^i = 0 mod p, for all 1 <= i <= p-1
\n" ); document.write( " Hence, (k+1)^p = E C(p,i) k^i = k^p + 1 mod p = (k + 1) mod p.
\n" ); document.write( " [ by the induction hypothesis]
\n" ); document.write( " This means (**) is true for a = k+1 and the inductive proof is complete.\r
\n" ); document.write( "\n" ); document.write( " This is an important fact about prime numbers . It is an direct result
\n" ); document.write( " about the group Zp.\r
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\n" ); document.write( "\n" ); document.write( " Kenny\r
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