If p is a prime then for any integer a

ap ≡ a (mod p)

ap ≡ 1 (mod p)

ap ≡ 0 (mod p)

ap ≡ -1 (mod p)

If p is a prime then for any integer a Correct Answer ap ≡ a (mod p)

Concept:

If p is a prime and a is an integer and if p doesn´t divide a, then a^p-1 ≡ 1(mod p).

Calculation:

Given, p is a prime and a is an integer

Then we have two cases

Case 1) p divides a

then a^p - a ≡ a(a^{p - 1} - 1)

⇒ p|(a^p - a)

⇒ a^p ≡ a (mod p)

Case 2) p doesn´t divide a

then a^p-1 ≡ 1(mod p)

Since a ≡ a (mod p)

we have,

a^{p - 1}. a ≡ 1.a (mod p)

⇒ a^p ≡ a (mod p)

Hence, the correct answer is option 1).

Feb 20, 2025

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