Fermat's Little Theorem definition clarification. Fermat's Little Theorem states that (acc to Gallian book)
$a^p \mod p= a \mod p$.
Does it mean that we get the same remainder when both $a^p$ and $a$ are divided by some prime $p$? I am quite confused about this statement. Through wikipedia,
I read $a^p \equiv a \mod p$. Kindly help. I am new to this number system topic. 

$a^p\mod p\equiv a \mod p\implies a^p-a\equiv 0 \mod p \implies p\text{divides} (a^p-a)\implies a^p-a=kp\implies a^p=kp+a$ 
So what is the remainder when $a^p$ is divided by $p$
