How many values can natural number n take if n! is a multiple of 76 but not 79?

June 23, 2019

how many values can natural number n take if n! is a multiple of 76 but not 79?

a) 7 b) 21 c) 14 d) 12

The smallest factorial that will be a multiple of 7 is 7!
14! will be a multiple of 72
Extending this logic, 42! will be a multiple of 76

However, 49! will be a multiple of 78 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 79.

In general for any natural number n,
n! will be a multiple of [n7]+[n49]+[n343]+ ………..
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 76. We also see that 56! is the smallest factorial that is a multiple of 79. So, n can take values {42, 43, 44, 45……..55}

There are 14 values that n can take.

Correct Answer: 14 values