Numbers that have exactly three factors?

Answer 1

Remember that two of the factors are ALWAYS 1 and the number itself. So you want just one more number. This means that the number has to be a perfect square:

4 -- 1, 2, 4

9 -- 1, 3, 9

25 -- 1, 5, 25

49 -- 1, 7, 49

You wouldn't choose 16 nor 36, because their square roots are not prime.

Answer 2

Actually, the square of any prime p has exactly 3
factors: 1, p and p².
In fact these are the only numbers with exactly 3 factors.
Why?
If p_1^a_1 ... p_k^a_k is the decomposition of n
into its prime factors,
the number of divisors of n is
(a_1+1)...(a_k+1)
If there are more than 2 prime factors of n, then
there are at least 4 factors of n.
So n has only 1 prime factor and if we
want n to have exactly 3 factors, it is clear
that we must have a_1 = 2.

Answer 3

The squares of a prime number have 3 divisors

Ana

Answer 4

4:1,2,4;
8:2,4,8;
9:1,3,9