2013 AMC 10B Problems/Problem 24

Problem

A positive integer $n$ is nice if there is a positive integer $m$ with exactly four positive divisors (including $1$ and $m$ ) such that the sum of the four divisors is equal to $n$ . How many numbers in the set $\{ 2010,2011,2012,\dotsc,2019 \}$ are nice?

$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$

Solution

A number with only four positive divisors has its prime factorization in the form of a*b, where a and b are both prime positive integers. The four factors of this number would be 1,a,b, and ab. The sum of these would be ab+a+b+1, which can be factored into the form (a+1)(b+1). Since all odd numbers have an odd number in their prime factorization, the product of (a+1)(b+1) results in an even number. Therefore we can rule all out all of the odd integers in the set. Considering the even cases, we see that 2016 is the only one that works. Thus the answer is (A) 1.

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2013 AMC 10B Problems/Problem 24

Problem

Solution