1. ## integers and divisors

(a) Determine which positive integers have exactly 3 positive divisors and prove your statement.
(b) Determine which positive integers have exactly 4 positive divisors and prove your statement.

2. Hello, smithhall!

I don't have rigorous proofs for my claims.

(a) Determine which positive integers have exactly 3 positive divisors

Suppose $\displaystyle N$ is the product of two distinct primes: .$\displaystyle N \:=\:p\cdot q$
. . Then $\displaystyle N$ has four divisors: .$\displaystyle 1,\: p,\: q,\: pq$

To have exactly 3 divisors, the two prime factors must be equal: .$\displaystyle N \:=\:p^2$
. . The divisors are: .$\displaystyle 1,\:p,\:p^2$

Therefore, N must be the square of a prime.

(b) Determine which positive integers have exactly 4 positive divisors
$\displaystyle N$ must be the product of exactly two distinct primes.
$\displaystyle N$ must be the product of exactly two distinct primes.
What about $\displaystyle N=p^3$?