What is the largest 2-digit prime factor of the integer $\binom{200}{100}$ [closed]

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

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2 Answers

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$\displaystyle\binom{200}{100}=\frac{200!}{100!~100!}$

every prime factor between 1 and 100 appears twice in the denominator.
So, we need to find a $2$–digit prime $p$ that appears three times in the numerator $200!$, or $3p< 200$.The largest such prime is $61$.

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Cute problem.

HINT: If $p$ is a prime, and $67\le p\le 100$, there are two factors of $p$ in the numerator and two factors of $p$ in the denominator of $$\binom{200}{100}=\frac{200!}{100!^2}\;.$$ What does it take to get another factor of $p$ in the numerator but not in the denominator?

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