M7MBA wrote:For any integer m greater than 1, $m denotes the product of all the integers from 1 to m, inclusive. How many prime numbers are there between $7 + 2 and $7 + 10, inclusive?

(A) None
(B) One
(C) Two
(D) Three
(E) Four

We see that $m is the conventional notation of m!. Thus, the problem asks for the number of prime numbers between 7! + 2 and 7! + 10, inclusive. Let's analyze each of these numbers.

7! + 2: Since 2 divides into 7! and 2, 7! + 2 has 2 as a factor, and thus it's not a prime.

7! + 3: Since 3 divides into 7! and 3, 7! + 3 has 3 as a factor, and thus it's not a prime.

7! + 4: Since 4 divides into 7! and 4, 7! + 4 has 4 as a factor, and thus it's not a prime.

7! + 5: Since 5 divides into 7! and 5, 7! + 5 has 5 as a factor, and thus it's not a prime.

7! + 6: Since 6 divides into 7! and 6, 7! + 6 has 6 as a factor, and thus it's not a prime.

7! + 7: Since 7 divides into 7! and 7, 7! + 7 has 7 as a factor, and thus it's not a prime.

7! + 8: Since 8 divides into 7! (notice that 7! has factors 2 and 4) and 8, 7! + 8 has 8 as a factor, and thus it's not a prime.

7! + 9: Since 9 divides into 7! (notice that 7! has factors 3 and 6) and 9, 7! + 9 has 9 as a factor, and thus it's not a prime.

7! + 10: Since 10 divides into 7! (notice that 7! has factors 2 and 5) and 10, 7! + 10 has 10 as a factor, and thus it's not a prime.

Thus, none of the integers between 7! + 2 and 7! + 10, inclusive, are prime.

Answer: A