Number Theory

Suppose the prime factorization of $m$ is $m = p_1^{k_1} \cdot p_2^{k_2} \cdots p_n^{k_n}$. Then the number of positive divisors is $(1 + k_1)(1 + k_2)\cdots(1 + k_n)$.

The ways of writing 8 into product of numbers are $8 = (1 + 1) \times (1 + 3) = (1 + 1) \times (1 + 1) \times (1 + 1)$. From this and the fact above, we know that $m$ is either of the form $pq^3$ or $pqr$, where p, q, r are distinct primes. The number of distinct prime factors of m is either 2 or 3. The given conditions are not sufficient to conclude definitely that the number of distinct prime factors is 2 (resp. 3).