Question

Wilson theorem states that if n is a prime number that n divides (n-1)! +1 using this find the smallest divisor of 12!+6!+12! × 6!+1!

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Answer 1

The answer is 7.

12! + 6! +12!x6! +1, can be written as

12! + 6!*12! + 6!+1

= 12!(6! +1) + 6!+1

= (12!+1)(6!+1)

So this number is divisible by 13 and 7. The smaller of the two is 7 so 7 must be the answer

Answer 2

Hey friend, Harish here.

Here is your answer:

$12!+ 6! +12! \times 6! +1!$

→   $12! \times 6!+12!+6!+1!$

→   $12!(6!+1)+1(6!+1)$

→   $(12!+1) ( 6!+1)$

These are the factor of the given equation . So the smallest divisor must also divide (6!+1) .

(6!+1) ≡ 0 mod 7 ( By using the  given wilson's theorem)

Therefore the smallest divisor is 7 if the asked a prime factor. But as they asked for the smallest divisor it is 1. 
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Hope my answer is  helpful to you.