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(C) To find the exponent of a prime $p$ in $n!$,we use Legendre's formula: $E_p(n!) = \sum_{k=1}^{\infty} \lfloor \frac{n}{p^k} \rfloor$.Here,$n = 100

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$48$

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(C) To find the exponent of a prime $p$ in $n!$,we use Legendre's formula: $E_p(n!) = \sum_{k=1}^{\infty} \lfloor \frac{n}{p^k} \rfloor$.Here,$n = 100$ and $p = 3$.$E_3(100!) = \lfloor \frac{100}{3} \rfloor + \lfloor \frac{100}{3^2} \rfloor + \lfloor \frac{100}{3^3} \rfloor + \lfloor \frac{100}{3^4} \rfloor$$E_3(100!) = \lfloor \frac{100}{3} \rfloor + \lfloor \frac{100}{9} \rfloor + \lfloor \frac{100}{27} \rfloor + \lfloor \frac{100}{81} \rfloor$$E_3(100!) = 33 + 11 + 3 + 1 = 48$.Thus,the exponent of $3$ in $100!$ is $48$.

The exponent of $3$ in $100!$ is

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