How many prime numbers exist between 260 and 280?

It's sometimes unfortunate that this site uses kudos to determine which posts rise to the top of a thread, because sometimes solutions that are incorrect end up featured, and can mislead readers. That's the case here.

It's true that prime numbers will be of the form 6n-1 or 6n+1, at least once you get beyond the smallest primes. But there's no *guarantee* that a number of the form 6n+1 or 6n-1 is prime. If the range in this question were different -- say it asked how many prime numbers there are between 240 and 260 -- then you'd think, just by finding numbers in the form 6n-1 and 6n+1, that 253 was prime (because it equals 6*42 + 1, so is in the form 6n+1). But it is not prime: 253 = 23*11. It's really just a lucky coincidence that the numbers from 260 to 280 that look like 6n-1 or 6n+1 are all prime in this question, once we rule out multiples of 5.

Unfortunately, very few of the solutions posted above are mathematically correct (the only correct solutions are the ones from sowragu and from nalinair, though neither fully explains the method they're using). We can easily rule out numbers between 260 and 280 that are not prime -- the even numbers are not, the numbers ending in '5' are not, and the numbers whose digits sum to a multiple of 3 are not. That leaves us with these candidates:

263, 269, 271, 277

But all we've proven so far is that the four numbers above are not divisible by 2, 3 or 5. We haven't come close to proving they're prime -- how do we know they aren't divisible by 7, say, or by 13? We have to check that. The most efficient way to prove a large number x is prime is to prove that it cannot be divided by any primes up to √x. If you can't divide x by any primes less than or equal to √x, then x is a prime number. So here, since √277 is just less than 17, to prove the numbers above are prime, we need to check if we can divide them by any prime less than 17. We've checked for 2, 3 and 5, but we still need to check for 7, 11 and 13. I'd do that by locating the easy multiples of 7, 11 and 13 that are nearby:

280 is a multiple of 7, so these are as well (subtracting 7s) : 273, 266, 259 --> none of our four candidates is a multiple of 7
220 is a multiple of 11, so these are as well (adding 44, 55 and 66): 264, 275, 286 --> none of our four candidates is a multiple of 11
260 is a multiple of 13, so these are as well (adding 13s): 273, 286 --> none of our four candidates is a multiple of 13

With all of that work done, we can safely conclude that the four candidates are all indeed prime numbers, so the answer is four.

Incidentally, to reliably test whether numbers this size are prime is a very time-consuming process, which is why no real GMAT question ever asks you to do it.