Find the number that divides 16!+1?

Wow, I was always confused with these question. Thanks

Yes, Wilson's theorem works for this particular number. Though you won't need this theorem for GMAT.

Using the logic => Multiple +non multiple = non multiple => discarding all the options => only 17 is the viable choose..
Remember => we dont really have to compute the values here.. All is in the logic

Bunuel could you advise me if the following is a correct solution or if I am inventing math?

16! = 17!/17 --> 17!/17 + 1 =(17!+17)/17 ---> 16!+1 divisible by 17

16!+1=(17!+17)/17 but how do you conclude that (17!+17)/17 is divisible by 17? In other words how do you know that [(17!+17)/17]/17 is an integer?

oh, yes I see, what I had in my mind does't actually make sense! thanks!

Learnt a new concept.
But when I apply the same to smaller factorials such as 5!+1 = 121 (divisible by 11 and not by 6-because of 3 & 2 in 5! but what about 7).

Is there any exception to the theorem?

You could see this theorem here: https://en.wikipedia.org/wiki/Wilson's_theorem

The theorem is applied for prime p and (p-1)!+1 divisible by p.

Hence, 4!+1 is divisible by 5. However, 5!+1 will not be divisible by 6.

This theorem could be proved by using advanced mathematic tools, thus this theorem is too hard and we no need to learn this theorem in solving GMAT PS/DS questions.

1) Two consecutive integers do not have any factors in common other than 1. This means that factor of 16!+1 has to be a number that is not a factor of 16!
2) 7, 1, and 6 are factors of 16! as they are multiples of 16!, 18 is also a factor of 16! because 2 and 9 are factors of 16!
3) By deduction, the only factor that is not a multiple of 16! is 17, and this means that it is a multiple of 16!+1

The correct answer is C.

We need to determine which one of the numbers in the given answer choices divides into 16! + 1. In other words, which one is a factor of 16!+1. To determine this, we must recognize that 16! and 16! + 1 are consecutive integers, and consecutive integers will never share the same prime factors. Thus, 16! and 16! + 1 must have different prime factors.

However, rather than breaking 16! factorial into primes, we can look at the answer choices to determine which choice is not a factor of 16!. Since 16! = 16 x 15 x 14…5 x 4 x 3 x 2 x 1, we see that choices A, B, and D are factors of 16! Since 18 = 2 x 9, 2 and 9 are also factors of 16!. However, none of these numbers (6, 7, 11, and 18) will be a factor of 16! + 1, so the only number that can be a factor of 16! + 1 is 17.

Answer: C

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