Question

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

• A. $33$
• B. $44$
• C. $48$
• D. $52$

Answer 1

The correct option is C

$48$

Step 1: Given data

The digit which exponent needs to be searched=$3$

The expression in which the exponent needs to be find=$100!$

Step 2: Formula used

We know that the exponent of a prime $p$ in $n!$ is given by $\sum _{r=1}^{\infty }\frac{n}{p^r}$

Step 2: Compute the number of exponents

So, the exponent $3$ in $100!$ will be,

$$\begin{gathered} =\frac{100}{3}+\frac{100}{3^2}+\frac{100}{3^3}+\frac{100}{3^4} \\ =33+11+3+1 \\ =48 \end{gathered}$$

So, the exponent of $3$ in $100!$ is $48$.

Hence, option (C) is the correct answer.