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Quantitative Aptitude

Highest Power of a Number

If 10000 !=10...

Question

If $10000! = (100!)^{K} \times P$ , where K and P are integers, then what is the maximum value of K?

A

103

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B

104

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C

105

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D

102

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Solution

The correct option is A 103
The key here are the prime numbers. K should be selected in such a way that it will depict the value of the highest power of the largest prime number before 100 which is a factor of 10000!. The largest prime number below 100 is 97 and the highest power of 97 in 10000! is 104.
The highest power of 2 in 100! is 97 and the highest power of $2^{97}$ in 10000! is 103.
This step needs to kept in mind.
Hence 103 is the highest power of 100! in 10000!.

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