It is confusing for students regarding the two forms of the Fermat’s Little Theorem (which is the generalization of the ancient Chinese Remainder Theorem):

**General: For any number a**

$latex boxed { a^p equiv a mod p, forall a}$

We get,

$latex a^{p} – a equiv 0 mod p$

$latex a. (a^{(p-1)} -1) equiv 0 mod p$

$latex p mid a.(a^{(p-1)} -1)$

If (**a**, p) co-prime, or g.c.d.(**a**, p)=1,

then p cannot divide **a**,

thus

$latex p mid (a^{(p-1)} -1)$

$latex a^{(p-1)} -1 equiv 0 mod p$

__Special: g.c.d. (a, p)=1 __

$latex boxed {a^{(p-1)} equiv 1 mod p, forall a text { co-prime p}}$