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I found some formula about special function very complicated, so I am curious how you people solve this by hand.

$\Gamma(a)+\Gamma(b)= 121\,645\,106\,635\,852\,800$

but $a$ and $b$ are very small actually, condition: at least one of $a$ and $b$ are integers.

Can we prove or disprove both of them must be integer?

Reference: http://www.wolframalpha.com/input/?i=gamma%2814%29%2Bgamma%2820%29

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    There are only $2 \ 0$s on the end so $a,b$ cannot both be >15, and neither is <11. The last digit non-zero digit of $6227020800$ is an $8$, and the digit before that is$a$$0$, and the last $2$ digits can easily be calculated to eliminate $10!,11!,12!,14!$.2012-04-21

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