\begin{align} 2\text{ goes into }100 & & 50\text{ times} \\ 2^2\text{ goes into }100 & & 25\text{ times} \\ 2^3\text{ goes into }100 & & 12\text{ times} \\ 2^4\text{ goes into }100 & & 6\text{ times} \\ 2^5\text{ goes into }100 & & 2\text{ times} \\ 2^6\text{ goes into }100 & & 1\text{ time} \\ \end{align} $ 50+25+12+6+2+1 = 96. $
Thus $2^{96}$ divides $100!$ and $2^{97}$ does not.
\begin{align} 5\text{ goes into }100 & & 20\text{ times} \\ 5^2\text{ goes into }100 & & 4\text{ times} \end{align}
Thus $5^{24}$ divides $100!$ and $5^{25}$ does not.
$\min\{96,24\}=24.$
So $(2\cdot5)^{24}$ divides $100!$ and $(2\cdot5)^{25}$ does not.