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?
Choose the correct option.
A) None  B) One  C) Two  D) Three  E) Four
$ is basically a factorial of a number.   So, we are asked to find the number of primes between 7!+2 and 7!+10, inclusive.   From each number 7!+k were 2≤k≤102≤k≤10 we can factor out k, thus there are no pries in the given range.   For example:   7!+2=2(3*4*5*6*7+1) --&gt; a multiple of 2, thus not a prime;   7!+3=3(2*4*5*6*7+1) --&gt; a multiple of 3, thus not a prime;   ...   7!+10=10(3*4*6*7+1) --&gt; a multiple of 10, thus not a prime.   Nonenswer: None.
<result>A</result>
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?