If $p$ and $q$ are distinct primes and $a$ be any integer then $a^{pq} -a^q -a^p +a$ is divisible by $pq$. If $p$ and $q$ are distinct primes and $a$ be any integer then $a^{pq} -a^q -a^p +a$ is divisible by $pq$.
Factorising we get $a^{pq} -a^q -a^p +a =a^p(a^q -1) - a(a^{q-1}-1)$ and we know $p \mid a^{p-1}-1$ and $q \mid a^{q-1}-1$.
I can't proceed further with the proof. Please Help!

For prime $q,$ 
$$a^{pq}-a^q-a^p+a=[(\underbrace{a^p})^q-(\underbrace{a^p})]-[a^q-a]$$
Now by Fermat's Little Theorem, $b^q\equiv b\pmod q$ where $b$ is any integer
Set $b=a^p, a$
Similarly for prime $p$
Now if $p,q$  both divides $a^{pq}-a^q-a^p+a,$ the later must be divisible by lcm$(p,q)$ 
