Number Theory

I'm not sure how to explain the prcoess, but the smallest number with 18 positive divisors is $180$

We have $1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180.$

Now, $180^2 = 32400$

You could also use prime factorization, by doing

$32400 =2 ^ 4 ×3 ^ 4 ×5 ^ 2$

$(4+1)(4+1)(2+1)=5×5×3=75$

Thus, our answeris 75. 

Thanks! :)

Depends on the factorization of m

m =  2 * 3^8   has  (1 + 1) (8 + 1)  = 18 divisors

m^2  = [ 2 * 3^8]^2  =  2^2 * 3^16  has ( 2 + 1) (16 + 1)  =  51 divisors

But

m = 2^2 * 3^5  has (2+1)(5 + 1)  = 18 divisors

m^2  = [ 2^2 * 3^5]^2  = 2^4 * 3^10  has (4 + 1) (10 + 1)  =  55 divisors

And as  NTS pointed out, 

m = 2^2 * 3^2 * 5  has (2 + 1) (2 + 1) (1 + 1)  =18 divisors

But

(2^2 * 3^2 * 5)^2 = 2^4 * 3^4 * 5^2   has ( 4 + 1 )(4 + 1) (2 + 1)  = 75 divisors