2007 iTest Problems/Problem 14

Problem

Let $\phi(n)$ be the number of positive integers $k< n$ which are relatively prime to $n$ . For how many distinct values of $n$ is $\phi(n)=12$ ?

$\text{(A) } 0 \quad \text{(B) } 1 \quad \text{(C) } 2 \quad \text{(D) } 3 \quad \text{(E) } 4 \quad \text{(F) } 5 \quad \text{(G) } 6 \quad \text{(H) } 7 \quad \text{(I) } 8 \quad \text{(J) } 9 \quad \text{(K) } 10 \quad \text{(L) } 11 \quad \text{(M) } 12\quad \text{(N) } 13\quad$

Solution

If $n=p_1^{k_1}\cdots p_s^{k_s}$ then $\phi(n)=p_1^{k_1-1}\cdots p_s^{k_s-1}\cdot (p_1-1)\cdots (p_s-1)$ . Hence every prime factor of $n$ is contained in $\{2,3,5,7,13\}$ . Let $p$ be the largest prime dividing $n$ . If $p=13$ then $n=13,26$ . If $p=7$ then $n=7k$ with $7\not |k, \phi(k)=2$ giving solutions $n=21,28,42$ . If $p=5$ then $n=5k$ with $5\not |k,\phi(k)=3$ , no solutions since $\phi(k)$ is always even unless it is 1. Otherwise $n=2^a\cdot 3^b$ where $a,b\ge 1$ and $\phi(n)=2^a\cdot 3^{b-1}$ . Hence $a=2,b=2,n=36$ . Therefore all solutions are given by $n=13,26,21,28,42,36$ and the answer is G.

~Solution by Alexheinis but not posted on the wiki by Alexhienis

2007 iTest (Problems, Answer Key)
Preceded by: Problem 13	Followed by: Problem 15
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2007 iTest Problems/Problem 14

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