Counting Divisors

If $n = p_1^{a_1} \times p_2^{a_2} \times ... \times p_k^{a_k}$, then the number of divisors is $(a_1 + 1)(a_2 + 1)...(a_k + 1)$.

For $72 = 2^3 \times 3^2$, the divisor count is $(3+1)(2+1) = 12$. Each divisor is formed by choosing a power of 2 (0 to 3) and a power of 3 (0 to 2). Product rule in action.