3. How many times does 24 divide into 100 ! (factorial)?
Solution: We first determine the number of times 2 and 3 divide into $100!=1 \cdot 2 \cdot 3 \cdots 100$. Let $\langle N\rangle_{n}$ be the number of times $n$ divides into $N$ (i.e. we want to find $\langle 100 \text { ! }\rangle_{24}$ ). Since 2 only divides into even integers, $\langle 100!\rangle_{2}=\langle 2 \cdot 4 \cdot 6 \cdots 100\rangle$. Factoring out 2 once from each of these multiples, we get that $\langle 100!\rangle_{2}=\left\langle 2^{50} \cdot 1 \cdot 2 \cdot 3 \cdots 50\right\rangle_{2}$. Repeating this process, we find that $\langle 100!\rangle_{2}=\left\langle 20^{50+25+12+6+3+1} \cdot 1\right\rangle_{2}=97$. Similarly, $\langle 100!\rangle_{3}=\left\langle 3^{33+11+3+1}\right\rangle_{3}=48$. Now $24=2^{3} \cdot 3$, so for each factor of 24 in 100! there needs to be three multiples of 2 and one multiple of 3 in 100 !. Thus $\langle 100 \text { ! }\rangle_{24}=\left(\left[\langle 100!\rangle_{2} / 3\right]+\langle 100 \text { ! }\rangle_{3}\right)=32$, where $[N]$ is the greatest integer less than or equal to $N$.