Question

Which of the following is a divisor of 12! · 6! + 12! + 6! + 1?

A. 21
B. 77
C. 91
D. 115
E. 143

Hint : Use Wilson's Theorem
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Answer 1

• C. 91

Explanation:

Rewriting the given as factors:

• 12! · 6! + 12! + 6! + 1 =
• 12!(6! + 1) + (6! + 1) =
• (12! + 1)(6! + 1) =
• ((13 - 1)! + 1)((7 - 1)! + 1)

As per Wilson's theorem, if n is prime number, (n - 1)! + 1 is divisible by n

As per above, 13 and 7 are both prime numbers, so the expression is divisible by both 7 and 13:

• 7*13 = 91

So correct choice is C