let $\displaystyle p_{i}$ be a prime number and $\displaystyle a_{i}$ be a positive integer for all $\displaystyle i$

a) list all distinct divisors of 32

1,2,4,8,16,32

$\displaystyle 2^5$ 5+1 = 6 divisors

b) how many distinct divisors does $\displaystyle p^a$ have?

a+1 divisors

c) how many distinct divisors does $\displaystyle p_{1}^{a_{1}}p_{2}^{a_{2}}\cdot\cdot\cdot p_{m}^{a_{m}}$ have?

$\displaystyle (a_{1}+1)(a_{2}+1)\cdot\cdot\cdot(a_{m}+1)$ divisors

d) how many distinct divisors does 60 have?

using prime factorization $\displaystyle 60 = 2^2\cdot3\cdot5$

so it has $\displaystyle (2+1)(1+1)(1+1) = 12$

here is the one I don't know how to solve

e) What is the smallest positive integer with exactly 100 distinct divisors?