10. (1 mark) How many positive integers less than 500 have exactly 15 positive integer factors?
(1 分) 小於 500 的正整數中有多少個剛好有 15 個正因子?
10. Since $15=1 \times 15=3 \times 15$ are the only factorizations of 15 , so a positive integer has exactly 15 positive integer factors if and only if it is of the form $p^{14}$ for some prime $p$, or $p^{2} q^{4}$ for two distinct primes $p$ and $q$. Now $2^{14}>500$, so there are no numbers of the first form between 1 and 500. The only one of the second form are $3^{2} \times 2^{4}=144,5^{2} \times 2^{4}=400$, and $2^{2} \times 3^{4}=324$, hence 3 is the answer.