Thread: euler theorem

Based on euler theorem, i know that if gcd(a,p)=1 then a^(p-1) congruent to 1 mod p. is (p-1) the smaller value such that this is true? Im wondering if it is possible that a^(q-1) is congruent to 1 mod p where (q-1)<p-1 and gcd(a,q)=1

It is certainly possible. Just take