Problem
NT-B1-M11-P012 Odd Number of Divisors
#12
★★☆☆☆ Level 2 of 5
Prove that a positive integer has an odd number of positive divisors if and only if it is a square.
Divisors usually pair as \(d\) and \(\frac{n}{d}\).
If \(d\mid n\), it pairs with \(\frac{n}{d}\). The two divisors are distinct except when \(d=\frac{n}{d}\), i.e. \(d^2=n\). Thus an unpaired divisor appears exactly for squares, and only then is the number of divisors odd.
Nice divisor-pairing idea.