1983 AIME Problems/Problem 8

Problem

What is the largest 2-digit prime factor of the integer ${200\choose 100}$ ?

Expanding the binomial coefficient, we get ${200 \choose 100}=\frac{200!}{100!100!}$ . Let the prime be $p$ ; then $10 \le p < 100$ . If $p > 50$ , then the factor of $p$ appears twice in the denominator. Thus, we need $p$ to appear as a factor three times in the numerator, or $3p<200$ . The largest such prime is $\boxed{061}$ , which is our answer.