I have no life

$20 = 2^2 5 \text{ so we'll want to use prime factors }\\ 2^4 \cdot 3^1 \cdot 5^1 = 240\\ \text{This has }(4+1)(1+1)(1+1)=20 \text{ divisors}$

I'm pretty sure for (2) we just find the smallest positive integer with 6 divisors and square it.

Proceeding as above

$6=2\cdot 3 \text{ so choose}\\ (2^2\cdot 3)^2 = 144$

a)

20 has factors of 1, 2, 4, 5, 10, and 20. So, to make the number as small as possible, we want to use 5 * 2 * 2. We will pair the largest exponents with the smaller primes to get 2^(5-1) * 3 * 5 = 240.

b)

It has 6 perfect square divisors, so we need the smallest number with 6 divisors then square that to get the answer. That is (by counting), 12. So, our answer is 12^2 = 144.