Problem
NT-B1-M01-P005 Odd Number of Divisors
#5
★☆☆☆☆ Level 1 of 5
Prove that a positive integer has an odd number of positive divisors if and only if it is a perfect square.
Divisors usually pair as \(d\) and \(\frac{n}{d}\).
If \(d\mid n\), then \(\frac nd\) is the paired divisor. Divisors come in pairs, except when \(d=\frac nd\), i.e. \(d^2=n\). Thus an unpaired divisor exists exactly when \(n\) is a square. Therefore the number of divisors is odd exactly for perfect squares.
A classical fact used in many later modules.