If N is a positive integer, does N have exactly three factors?

In order to have three factors exactly, N must be a square. Most numbers have an even number of factors because factors come in pairs, take 12= 1*12 = 2*6=3*4 for example. Only squares have an odd number of factors.

Statement 1:

First of all \(N^2\) must have factors 1, N, and \(N^2\). N cannot be prime or else we're stuck with 3 factors only. If N was a product of 2 different prime numbers, we would have N = a*b, and \(N = a^2*b^2\) which would give (2+1)(2+1) = 9 factors. Finally, if N was a square of a prime, \(N = a^2\) would mean the factors are 1, a, \(a^2\), \(a^3\), and \(N^2 = a^4\) which is exactly 5 factors.

Thus N is a square of a prime, and N must have exactly three factors, 1, a, and N. Sufficient.

Statement 2:

If N is prime then N has only 2 factors. N could be a square of a prime which is 3 factors. Insufficient.

Ans: A

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