12. Consider any set of 20 consecutive integers, all greater than 50. What is the maximum number of them that can be prime numbers?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8 .

## Numerical Answer Problems - 8 points
12. The answer is $\mathbf{( C )}$. We can first exclude all even numbers from the possible prime numbers, and thus consider only sets of 10 consecutive odd numbers greater than 50. Among them, at least three are multiples of 3 and exactly two are multiples of 5. The two numbers that are multiples of 5 differ by 10, so at most one of them is also a multiple of 3. It follows that there are at least $3+2-1=4$ odd numbers that are multiples of either 3 or 5, and therefore are not prime. The total number of possible prime numbers is thus less than or equal to 6.

The number we are looking for is, however, exactly equal to 6, as shown by the example of the 6 prime numbers $97,101,103,107,109,113$ in the interval $\{96, \ldots, 115\}$.