What is the smallest positive integer $ n $ for which $\frac{50!}{24^n}$ is not an integer?

Question

We can go for a direct check! But it is too tedious! Is there any result that can be applied to find the answer.

Answer 1

As $24=3\cdot2^3$

Using Highest power of a prime $p$ dividing $N!$,

the highest power of $2$ that divides $50!$ is $\displaystyle\sum_{r=1}^\infty{\left\lfloor\frac{50}{2^r}\right\rfloor}=25+12+6+3+1=47$

the highest power of $8=2^3$ that divides $50!$ is $\displaystyle\left\lfloor\frac{47}3\right\rfloor=15$

Similarly, the highest power of $3$ that divides $50!$ will be $22$

So, the highest power of $24$ that divides $50!$ will be min$(22,15)$