Problem
NT-B2-M10-P008 When \(\varphi(n)=2n/5\)
#8
★★★☆☆ Level 3 of 5
Find all positive \(n\) such that \(\varphi(n)=\frac{2n}{5}\).
1001 Problems in Classical Number Theory (method inspiration) · Problem 732
Equivalently, \(\frac{n}{\varphi(n)}=\frac{5}{2}\).
We need \(\prod_{p\mid n}\frac{p}{p-1}=\frac52\). The factors from \(2\) and \(5\) give \(2\cdot\frac54=\frac52\). Adding any other prime changes the product. Without \(2\) or without \(5\), the value is also impossible. Therefore \(n=2^a5^b\), where \(a,b\ge 1\). Check: \(\varphi(2^a5^b)=2^{a-1}\cdot 4\cdot 5^{b-1}=\frac{2}{5}2^a5^b\).
Trains work with a rational ratio.