Question

Find the exponent of $3$ in $100!$.

Answer 1

In terms of prime factors $100!$ can be written as $2^a{.3}^b{.5}^c{.7}^d....$
Now, $E_3(100!)=\left[\frac{100}{3}\right]+\left[\frac{100}{3^2}\right]+\left[\frac{100}{3^3}\right]+\left[\frac{100}{3^4}\right]$
$=33+11+3+1=48$
Hence, the exponent of $3$ in $100!$ is $48$ ($\because 100!=2^a{.3}^b{.5}^c{.7}^d$ )