+167 Let $m$ be a positive integer such that $m$ has exactly $8$ positive divisors. How many distinct prime factors does $m$ have?
+9673 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).
