integers and divisors

**smithhall** · April 26th 2009, 10:03 PM

(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.

**Soroban** · April 27th 2009, 02:03 AM

Hello, smithhall!

I don't have rigorous proofs for my claims.

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

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

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

Therefore, N must be the square of a prime.

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

We answered this question in part (a).

$N$ must be the product of exactly two distinct primes.