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# Number Theory

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Find the largest prime divisor of 11! + 12! + 13!.

May 1, 2022

#1
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It is 13. Because a prime number has 0 factors, it can't be any bigger.

May 1, 2022
#2
+9459
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We factorize it first.

$$11! + 12! + 13! = 11! (1 + 12 + 12 \times 13) = 13^2 \times 11!$$

All prime factors of $$11!$$ are smaller than or equal to 11 since $$11! = 11 \times 10 \times 9 \times \cdots \times 2\times 1$$.

The largest prime factor is 13.

May 2, 2022