All exclamation marks - Part 3

Number Theory Level 2

What is the integer value of $n> 1$ for which $(n-1)! + 2$ is divisible by $n$ ?

See Part 1 and Part 2 .

The answer is 4.

1 solution

Amir Nisar
Jun 13, 2015

when we take n=4 then ( 4 - 1) ! +2 = 3! + 2 = 6 +2 =8 . which is divisible by 4.

Moderator note:

Your solution is incomplete. Even after getting the answer right, you should always consider alternative solutions.

By Wilson's Theorem , the only possible whole number $n$ that satisfy the congruence $(n-1)! \equiv 2 \pmod n$ is $4$ .

Is there a proper mathematical arithmetic solution, or is this just solve by guess and check?

Kelvin Khorn - 6 years ago

kelvin, the solution is not based on guess. There is a proper arithmetic solution. Before solving this question , one must understand the concept of factorial notation (! ). For example , when we write 5! this means 5 .4 .3.2.1=120. In next , we have to focus on the understanding of question that demands the value of n > 1. Greater values are 2,3,4,5,6,7,....................... . but we have to check here that the value on "n " should be greater than one and the result which is obtained through ( n- 1) ! +2 must be divisible by divisible by n. i hope you got the answer .

amir nisar - 5 years, 12 months ago

Thanks Amir Nisar, i get it now much appreciated!

Kelvin Khorn - 5 years, 11 months ago

All exclamation marks - Part 3

The answer is 4.

1 solution

Moderator note:

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