document.write( "Question 1019197: An integer n is called square-free if there does not exist a prime number p such that $\"p%5E2\"$ divides n.
\n" ); document.write( "a) Let $\"n%3E=2\"$ be an integer with the property that $\"a%5En=a++%28mod+n%29\"$ for every integer a. Prove that n is square-free.
\n" ); document.write( "b) Give an example of a square-free integer $\"n%3C=30\"$ such that $\"a%5En\"$ != a (mod n) for some integer a. \n" ); document.write( "

Algebra.Com's Answer #635140 by richard1234(7193) $\"\"$ $\"About$

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a) Suppose n is divisible by

for some prime p. Because the statement must hold for every integer a, it must hold when a = p. However the congruence

cannot possibly hold because

is divisible by

, and if the congruence holds mod n, it must hold mod

. So we have a contradiction, so n is square-free.\r
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\n" ); document.write( "\n" ); document.write( "b) If n is prime, then the congruence always holds by Fermat's little theorem. So we should pick a composite number, say n = 4. The congruence fails to hold true when a = 2, since

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