Bunuel wrote:

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

(1) The integer N^2 has exactly five factors

(2) Only one factor of N is a prime number

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:This is a question about prime numbers in disguise. Of course, any prime number has exactly two factors: 1 and itself. If we multiple two different primes, say 2 and 5, we get a number with four factors: the factors of 10 are {1, 2, 5, 10}. The only way to get a number with exactly three factors is if the number is the perfect square of a prime number. For example, 9 is the square of 3, and the factors of 9 are {1, 3, 9}; 25 is the square of 5, and the factors of 25 are {1, 5, 25}. If P is a prime number, then the factors of P squared are (a) 1, (b) P, and (c) P squared. Three factors. That’s what the question is asking. Is N the perfect square of a prime number.

Statement #1: very interesting.

If N is a prime number itself, then its square only has three factors. Squaring a prime doesn’t produce enough factors. This doesn’t meet the condition of this statement.

If N is the product of two primes, then its square has 7 factors. For example, 2*5 = 10, and 10 square is 100 which has seven factors: {1, 2, 4, 5, 10, 20, 25, 100}. Squaring a product of primes produces too many factors. This also doesn’t meet the condition of this statement.

The only way the square of N could have five factors is if N is the square of a prime number. Suppose N = 2 square, which is 4. Then N squared would be 16, which has five factors: {1, 2, 4, 8, 16}. In general, if P is a prime number, and N equals P squared, then N squared would equal P to the 4th power, which has five factors:

{1, P, P^2, P^3, p^4}

Therefore, N must be the square of a prime number, so we can give a clear “yes” answer to the prompt question. This statement, alone and by itself, is sufficient.

Statement #2:This is tricky. If we know that N is the square of a prime number, then this statement would be true, but that’s backwards logic. We want to know: if this statement is true, does it allow us to conclude that N is the square of a prime number? If only factor of N is a prime number, then N could be:

(a) a prime number: the only prime factor of 7 is 7.

(b) any power of a prime number: the only prime factor of 7 to the 20th is 7

So N could be the square of a prime number, or just the prime number itself, or the cube of the prime number, or the fourth power, or etc. Any number of factors would be possible, so we have no way to answer the question. This statement, alone and by itself, is not sufficient.

Answer = (A)

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