Problem
NT-B1-M09-P013 Odd Number of Divisors
#13
★★★☆☆ Level 3 of 5
Prove that a positive integer has an odd number of positive divisors if and only if it is a square.
Pair divisors as \(d\) and \(\frac{n}{d}\).
If \(d\mid n\), then \(\frac{n}{d}\mid n\). Divisors come in pairs except when \(d=\frac{n}{d}\), i.e. \(d^2=n\). Thus an unpaired divisor exists exactly when \(n\) is a square. Hence the number of divisors is odd exactly for squares.
One of the key theorems of the module.