N-is-completely-divisible-by-13-52-What-is-the-sum-of-the-digits-of-the-smallest-such-number-N-

Question Number 113001 by Aina Samuel Temidayo last updated on 10/Sep/20

N! is completely divisible by 13^{52} .

What is the sum of the digits of the

smallest such number N ?

Commented by MJS_new last updated on 11/Sep/20

N! = 13^{52} \times k

13! has 1 factor 13

(2 \times 13)! has 2

(13 k)! has k for k < 13

(13^{2})! has 14

(13 k)! has k + 1 for k < 26

(13 \times 26)! has 28

(13 k)! has k + 2 for k < 39

(13 \times 39)! has 42

(13 k)! has k + 3 for k < 52

\Rightarrow (13 \times 49)! id divisible by 13^{52}

but I {can}^{'} t calculate the sum of the digits of 637!

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