I know the formula to calculate this, but I don't understand the reasoning behind it:
For example, the number of trailing zeros in $100!$ in base $16$:
$16=2^4$,
We have: $\frac{100}{2}+\frac{100}{4}+\frac{100}{8}+\frac{100}{16}+\frac{100}{32}+\frac{100}{64}=97$
Number of trailing zeros$ =\frac{97}{4} = 24$.
Why do we divide by the power of '$2$' at the end?