Problem
NT-B2-M10-P007 When \(\varphi(n)=n/3\)
#7
★★★☆☆ Level 3 of 5
Find all positive \(n\) such that \(\varphi(n)=\frac{n}{3}\).
1001 Problems in Classical Number Theory (method inspiration) · Problem 733
You need \(\frac{n}{\varphi(n)}=3\).
We have \(\frac{n}{\varphi(n)}=\prod_{p\mid n}\frac{p}{p-1}=3\). To get the integer value \(3\), the primes \(2\) and \(3\) must occur: \(\frac{2}{1}\cdot\frac{3}{2}=3\). Any other prime \(p\ge 5\) adds a factor \(\frac{p}{p-1}>1\) and changes the value. Hence \(n=2^a3^b\), where \(a,b\ge 1\). All such \(n\) work.
A good next step after \(\varphi(n)=n/2\).