+367 a)What is the smallest positive integer that has exactly 20 positive divisors?
b)What is the smallest positive integer that has exactly 6 perfect square divisors?
+6248 \(20 = 2^2 5 \text{ so we'll want to use prime factors }\\ 2^4 \cdot 3^1 \cdot 5^1 = 240\\ \text{This has }(4+1)(1+1)(1+1)=20 \text{ divisors}\)
I'm pretty sure for (2) we just find the smallest positive integer with 6 divisors and square it.
Proceeding as above
\(6=2\cdot 3 \text{ so choose}\\ (2^2\cdot 3)^2 = 144\)
+532 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.
