Find the highest power of 5625 that can exactly divide 565!. Options: A . &nbsp 34 B . &nbsp 35 C . &nbsp 32 D . &nbsp 30

Find the highest power of 5625 that can exactly divide 565!.

Question

Find the highest power of 5625 that can exactly divide 565!.

Options:

A . 34

B . 35

C . 32

D . 30

Answer: Option A
:
A

$5625 = 5^{4} \times 3^{2}$ .
$∴$ Highest power of 5625 in 565! will depend on the highest power of $5^{4}$ .
So, highest power of $5^{4}$ in 565! is = Integral value of $\frac{113 + 22 + 4}{4}$ = 34.

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