Problem
NT-B2-M10-P004 Parity of Euler's Function
#4
★★☆☆☆ Level 2 of 5
Prove that for \(n>2\), the number \(\varphi(n)\) is even.
1001 Problems in Classical Number Theory (method inspiration) · Problem 523
Pair every coprime residue \(a\) with \(n-a\).
If \(\gcd(a,n)=1\), then \(\gcd(n-a,n)=1\). These two residues are distinct: \(a=n-a\) would give \(2a=n\), so \(a\) is not coprime to \(n\) for \(n>2\). Thus the coprime residues split into pairs, and their number \(\varphi(n)\) is even.
A standard idea; useful before problems involving \(\varphi(n)\).