AT-OEMO-FED1 2004 Problem 3

For natural numbers $a, b$ , define $Z(a,b)=\frac{(3a)!\cdot (4b)!}{a!^4 \cdot b!^3}$ .(a) Prove that $Z(a, b)$ is an integer for $a \leq b$ .(b) Prove that for each natural number $b$ there are infinitely many natural numbers a such that $Z(a, b)$ is not an integer.[/list]