Use Wilson Theorem for the divisor

Question

Use Wilson Theorem to find the smallest possible number which completely divides (12! + 6! + 12! × 6! + 1!).

Wilson Theorem →

Wilson Theorem states that if n is a prime number then n divides [(n-1)!+1] completely.

↓↓ Answer Below ↓↓

. . . . . . .

The answer is 7

Answer 1

By Wilson's Theorem, $7$ divides $(7-1)!+1 = 6!+1$

$(12! + 6! + 12! × 6! + 1!) = (12! * 6! + 12!) + (6! +1) = (6!+1) * (12!+ 1)$, meaning this giant number will have prime factors $7$ and $13$, because $7$ is the smaller of the two, our answer must be $7$.

Answer 2

1 is trivial

$2,3,4,5,6$ divide $6!$ and $12!$ therefore cannot divide $12! + 6! + 12!\times 6! + 1$ as each would divide the first three terms but not the last.

$7$ divides $12!$ and $12!\times 6!$ and by Wilson's theorem, outlined above, divides $6! + 1 $