The smallest factorial that will be a multiple of 7 is 7!
14! will be a multiple of 7^{2}
Extending this logic, 42! will be a multiple of 7^{6}
However, 49! will be a multiple of 7^{8} as 49 (7 * 7) will contribute two 7s to the factorial. (This is a standard question whenever factorials are discussed). Extending beyond this, 56! will be a multiple of 7^{9}.
In general for any natural number n,
n! will be a multiple of + ...........
where [x] is the greatest integer less than or equal to x. A more detailed discussion of this is available on this link
So, we see than 42! is a multiple of 7^{6}. We also see that 56! is the smallest factorial that is a multiple of 7^{9}. So, n can take values {42, 43, 44, 45........55}
There are 14 values that n can take.
Correct Answer: 14 values