If y is a positive integer, is y prime?

A prime number is a number whose factors a 1 and that number alone. For example, the number 11 is a prime number since its factors are only the numbers 1 and 11 (i.e. itself), whereas the number 12 is not a prime number since it has other factors besides 1 and 12 (e.g. 2, 3, 4, and 6).

(1) y>4!
This tells us that y is any number greater than 4!. Since the numbers greater than 4! include both prime and non-prime numbers, this statement is insufficient.

(2) y = 11!-12
This statement gives us the precise value of y (since we can calculate the numerical value of 11!-12), and thus determine whether y is prime. Statement (2) is therefore sufficient.

The answer is thus (B).

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by Ian Stewart » Sat Mar 02, 2019 4:39 am

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This is a near-exact replica of an official question. Statement 1 is clearly not sufficient, because there is an infinite number of primes. I think Statement 2 is supposed to read:

11! - 12 < y < 11! - 2

That's also not sufficient, because each number in that interval has an obvious divisor. Take a random number in that range, say, 11! - 7. Well, 11! alone is divisible by 7, so 11! - 7 must also be divisible by 7 (we can factor out a 7 from each of the two numbers in that subtraction). The same is true for any other number in that interval, so there are no primes between 11! - 12 and 11! - 2, and we have enough information to give a 'no' answer to the question.

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