Bunuel wrote:
If n is a positive integer, does n have four or more distinct factors?
(1) n is not prime
(2) 150 ≤ n < 200
Kudos for a correct solution.
MAGOOSH OFFICIAL SOLUTION:Keep in mind, a prime number is number with only two factors, 1 and itself. All positive numbers with only two positive factors are prime numbers.
If we multiple one prime by another prime, say 2*5 = 10, then we get four factors, {1, 2, 5, 10}. Any product of two different primes has four factors. If one of the numbers we multiply is not prime, then it already has more than two factors of its own, and so the resultant product will have many more than 4 factors. It would seem that non-prime numbers have to have at least 4 factors.
The one exception is: squares of prime numbers. Consider what happens when we square 5: we get 25, and the factors of 25 are {1, 5, 25}. The square of a prime number is not prime, and it has three factors.
Statement #1: n is not prime
Well, if n is not prime, n could be 24 or 25. The first, 24, has factors {1, 2, 3, 4, 6, 8, 12, 24}—eight factors, so the answer to the prompt question is “yes.” Meanwhile, 25 has three factors, so the answer to the prompt question is “no.” Two different answers to the prompt are possible. This statement, alone and by itself, is not sufficient.
Statement #2: 150 ≤ n < 200
Clearly, many of the numbers in that range will have more than 4 factors, and the answer to the prompt is “yes.” But here, n could be prime, and there must be some prime numbers in that range. You don’t need to know this, but the prime numbers in this range are {151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199}. If n were any of these numbers, n would have only two factors, and the answer to the prompt question would be “no.” Two different answers to the prompt are possible. This statement, alone and by itself, is not sufficient.
Combined statements:
Now, n must be in the 150 to 200 range, and all prime numbers are excluded. As note, most of the numbers in this range will have a large number of factors, so for almost all of them, the answer to the prompt question is “yes.” The trouble is: there one number in this range that is the square of a prime number. The square of 13 is 169, so the factors of 169 are {1, 13, 169}. Since this is the square of a prime, it has exactly three factors. This gives a “no” answer to the prompt question. Even with combined statements, we still can get two different answers. Even combined, the statements are not sufficient.
Answer = (E)
This explanation was incredibly useful especially to apply to future questions, thanks a lot!