# CAT Practice : Number System: Factorial

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Finding the highest power of 7 that divides n! is easy. Find the smallest n such that n! is a multiple of 7^32 might not be easy. Think about that...

## Factorials - basic

Q.3: How many values can natural number n take, if n! is a multiple of 76 but not 79?
1. 7
2. 21
3. 14
4. 12

Choice C. 14

## Detailed Solution

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 $\left[ {{{\rm{n}} \over {\rm{7}}}} \right]{\rm{ + }}\left[ {{{\rm{n}} \over {{\rm{49}}}}} \right]{\rm{ + }}\left[ {{{\rm{n}} \over {{\rm{343}}}}} \right]$+ ...........
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.

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## More questions from Number System - Factorial

This idea is so good that it comes with an exclamation mark. N! holds marvels that you might not have noticed before. Enter here to see those.