This is my working

$p^2-q^2=(p+q)(p-q)$

As both the integers on right hand side are even .It is divisible by 4.

Primes are of form either $3k+1$ or $3k+2$ .we get 3 possible cases for for $p$ and $ q$ After substituting them we notice that it is also divisible by 3. It is divisible by both 4 and 3. Therefore it is also divisible by 12.But how to show that it is divisible by 24