Abstract Algebra

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Powers of Integers mod p

ABSALG-KIN4EH

Which of the following statements is TRUE?

A

$a^{p-1}\equiv a\;{\rm mod}\; p$, for all integers $a$, and all primes $p>1$.

B

$a^{p-1}\equiv 1\;{\rm mod}\; p$, for all integers $a$, and all primes $p>1$.

C

For every integer $a$ coprime to $7$, we have
$a^6\equiv a^{12}\;{\rm mod}\;7$.

D

The set of residues mod $7$ of the numbers $r=1,2,\ldots,6$ is distinct from the set of residues
mod $7$ of the numbers $5r$, $r=1,2,\ldots,6$.

E

The set of residues mod $10$ of the numbers $r=1,3,7,9$ is distinct from the set of residues mod $10$ of
the numbers $3r$, $r=1,3,7,9$.