What is the largest $2$-digit [prime](https://artofproblemsolving.com/wiki/index.php/Prime) factor of the integer $n = {200\choose 100}$?
Solution 1
Expanding the [binomial coefficient](https://artofproblemsolving.com/wiki/index.php/Combination), we get ${200 \choose 100}=\frac{200!}{100!100!}$. Let the required prime be $p$; then $10 \le p  50$, then the factor of $p$ appears twice in the denominator. Thus, we need $p$ to appear as a factor at least three times in the numerator, so $3p3\]
Since $p<\frac{200}{3}=66.66...$, the largest prime value for $p$ is $p=\boxed{61}$
~ Nafer