The sum of prime numbers that are greater than 50 but less than 60 is

Official Explanation

The key to this question is that we are only talking about 9 numbers that are possibly prime. A formal mathematical approach would be quite difficult, but analysis by cases is easy if we know a few basic rules. There is a finite list of possibilities to consider, so we can start by considering each of these numbers and determining whether or not it's prime.

So, we will start with 51 and count up. The number 51 is divisible by 3, because its digits sum to 6, which is divisible by 3. Then, 52 is even, and the only even prime number is 2, so 52 is not a prime number. The number 53 might be prime. A sure way to test whether a number is prime is to start with a nearby perfect square. 53 is less than 64, which is 8^2. That means we can find out whether 53 is prime by checking all integers from 1 to 8. The reason is this works is that factors come in pairs, so if 53 had a factor above 8, it would be paired with a factor below 8. The number 2 isn't a factor of 53, because 53 is odd. The number 3 isn't a factor, because, the digits of 53 add up to 8, which is not divisible by 3. The number 4, 5, and 6 all also fail to go into 53. You can check each one either by doing long division, or by counting out multiples. For example, 4*12 and 4*13 and 4*14, so 4 is not a factor of 56. Similarly, 7 and 8 are not factors of 53 because they multiply to 56, which is only 3 away. Therefore, 53 is a prime number: its only factors are 1 and itself.

The next few numbers are not prime. Namely, 54 is even, 55 is divisible by 5, 56 is even. The number 57 has digits that sum to 12, which is divisible by 3, so 57 is divisible by 3. Then, 58 is even, so it's not prime. The number 59 might be prime. So for primes, we have 53 and maybe 59. Checking the answer choices, I see that 53 isn't an option, so 59 must be a prime (which we could have confirmed), and the correct answer is 53 + 59 = 112, or (B).

No GMAT question will require you to have memorized primes, but you have to know how to identify whether a number is prime, using the method we just discussed. The method is not much harder for larger numbers. For example, say, on a different question, you want to determine whether the number 111 is prime. The closest perfect squares are 10^2 and 11^2 = 121, so to find out whether 111 is prime, we will have to check at most possible factors from 2 up to 10. As it turns out, in this case we will stop at 3. We can see that 111 is divisible by 3 either by performing long division or by seeing that the digits of 111 sum to a number that is divisible by 3 (in this case, they sum to 3).

Again, the correct answer is (B).