Track / Stage 5 / 44 of 400 #644 of 2000

Problem 644

AIME late

Number theory Difficulty 5.1 Find the answer

9) If $n$ is a natural number with 6 positive integer divisors, how many positive integer divisors does $n^{2}$ have? N.B.: among the divisors of a number, we also count 1 and the number itself.
(A) 11
(B) 12
(C) 15
(D) 36
(E) the answer depends on $n$

Your answer

Official solution

9. The answer is $(\mathbf{E})$ .

Consider the number $n=2^{5}=32$ which has exactly 6 divisors: $1,2,4,8,16,32$ . Its square $n^{2}=2^{10}=1024$ has exactly 11 divisors, namely all numbers of the form $2^{k}$ , with $k=0,1,2, \ldots, 10$ . This example rules out answers (B), (C), and (D). On the other hand, if we consider $n=12$ , we see that it also has exactly 6 divisors: 1, 2, 3, 4, 6, 12; its square $n^{2}=144=2^{4} \cdot 3^{2}$ has as divisors all numbers of the form $2^{k} \cdot 3^{h}$ with $k=0,1,2,3,4$ and $h=0,1,2$ , which are a total of 15. Therefore, answer (A) is not correct, and we see that the number of divisors of $n^{2}$ in general depends on $n$ .

[Problem proposed by A. Bianchi]

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.