Derive the congruence: a^13 congruent a(mod 3*7*13) for all a?

Answer 1

Look at each modulus separately:
a^13 = (a²)^6 *a = a(mod 3),
because a² = 1(mod 3) for any a by Fermat's
little theorem.
a^13 = (a^6)² *a = a(mod 7),
because a^6 = 1(mod 7), by Fermat's
little theorem.
Finally, a^13 = a(mod 13), again by Fermat's
little theorem.
So, by the Chinese remainder theorem,
a^13 = a(mod 3*7*13) for all a.