# 1996 AHSME Problems/Problem 29

## Problem

If $n$ is a positive integer such that $2n$ has $28$ positive divisors and $3n$ has $30$ positive divisors, then how many positive divisors does $6n$ have? $\text{(A)}\ 32\qquad\text{(B)}\ 34\qquad\text{(C)}\ 35\qquad\text{(D)}\ 36\qquad\text{(E)}\ 38$

## Solution

Working with the second part of the problem first, we know that $3n$ has $30$ divisors. We try to find the various possible prime factorizations of $3n$ by splitting $30$ into various products of $1, 2$ or $3$ integers. $30 \rightarrow p^{29}$ $2 \cdot 15 \rightarrow pq^{14}$ $3\cdot 10 \rightarrow p^2q^9$

$5\cdot 6 \rightrarrow p^4q^5$ (Error compiling LaTeX. ! Undefined control sequence.) $2\cdot 3\cdot 5 \rightarrow pq^2r^4$

The variables $p, q, r$ are different prime factors, and one of them must be $3$. We now try to count the factors of $2n$, to see which prime factorization is correct and has $28$ factors.

In the first case, $p=3$ is the only possibility. This gives $2n = 2\cdot p^{28}$, which has $2\cdot {29}$ factors, which is way too many.

In the second case, $p=3$ gives $2n = 2q^{14}$. If $q=2$, then there are $16$ factors, while if $q\neq 2$, there are $2\cdot 15 = 30$ factors.

In the second case, $q=3$ gives $2n = 2p3^{13}$. If $p=2$, then there are $3\cdot 13$ factors, while if $p\neq 2$, there are $2\cdot 2 \cdot 13$ factors.

In the third case, $p=3$ gives $2n = 2\cdot 3\cdot q^9$. If $q=2$, then there are $11\cdot 2 = 22$ factors, while if $q \neq 2$, there are $2\cdot 2\cdot 10$ factors.

In the third case, $q=3$ gives $2n = 2\cdot p^2\cdot 3^8$. If $p=2$, then there are $4\cdot 9$ factors, while if $p \neq 2$, there are 2\cdot 3\cdot 9$factors. In the fourth case,$ (Error compiling LaTeX. ! Missing $inserted.)p=3 $gives$2n = 2\cdot 3^3\cdot q^5 $. If$q=2 $, then there ar$7\cdot 4= 28$factors. This is the factorization we want.

Thus,$(Error compiling LaTeX. ! Missing$ inserted.)3n = 3^4 \cdot 2^5 $, which has$5\cdot 6 = 30 $factors, and$2n = 3^3 \cdot 2^6 $, which has$4\cdot 7 = 28$factors. In this case,$ (Error compiling LaTeX. ! Missing $inserted.)6n = 3^4\cdot 2^6 $, which has$5\cdot 7 = 35 $factors, and the answer is$\boxed{C}$