Let n has an odd number of divisors

then this has 1+prime factors each to even power.

If n consists of 1+prime factors each to an even power

it must be a perfect square.

out of odd number of factors of the number N,

1 is common factor rest of the factors can be taken in pairs

like say x,y,z are rest of the factors ... then N/x , N/y, N/z are also the factors as pair which can be expressed as prime factors each to an even power

and suppose that x factor have no pair then x and N/x is unique that is x=N/x

that is N=x^2

that is perfect square.