Question

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

• A. 103
• B. 104
• C. 105
• D. 102
• E. 100

Answer 1

The correct option is A 103

option (A)

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.
$\to$ But 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!