To find the maximum power of a number which divides a factorial number, we need to consider how many of these numbers contained in the factorial.
Solved Example 1:
The maximum power of 5 in 60!
60! = 1 x 2 x 3 ..................60 so every fifth number is a multiple of 5. So there must be 60/5 = 12
In addition to this 25 and 50 contribute another two 5's. so total number is 12 + 2 = 14
Here [ ] Indicates greatest integer function.
Divide 60 by 5 and write quotient. Omit any remainders. Again divide the quotient by 5. Omit any remainder. Follow the procedure, till the quotient not divisible further. Add all the numbers below the given number. The result is the answer.
Solved Example 2:
Find the maximum power of 15 in 100!
We should commit to the memory that the above method is applicable only to prime numbers. But 15 is a composite number. (15 = 3 x 5). So we find the maximum power of 3 and 5 in the above expression.
Important note: The maximum power of 5 in the above expression is less than the maximum power of 3 as 5 is bigger number.