Divisibility

Let $p_{1}, p_{2}$ and $p_{3}$ be three distinct primes. How many positive divisors does the integer $p_1^{2} \cdot p_2^{4} \cdot p_3$ have?

(Hint: think about representing each divisor as a product of primes. Can there be any primes distinct from $p_{1}, p_{2}$ and $p_{3}$ in such a representation? Can this representation contain, for example, $p_1^{3}$?)