To prove that a^5 ≡ a (mod 15) for any integer a, it is necessary to demonstrate that a^5 - a is divisible by 15. The expression can be factored as (a-1)a(a+1)(a^2 + 1). To establish divisibility by 15, it must be shown that this expression is divisible by both 3 and 5. Divisibility by 3 can be confirmed since among any three consecutive integers (a-1, a, a+1), at least one is divisible by 3. Similarly, among any five consecutive integers, at least one is divisible by 5, ensuring the entire expression is divisible by 15.
