OG11 Question 153

cliou21 wrote:Can someone please explain this better then the OG?

153. Does the integer k have a factor p such that 1>p>k?
(1) k>4!
(2) 13! + 2 <= k <= 13! + 13

I understand that (1) is not sufficient.
With (2), how can you be sure that k is not a prime number between 13!+2 and 13! + 13?

Thanks!

K will never be a prime number if it is an integer between 13!+2 and 13!+13. Here is why. 13! (= 1*2*3*...*12*13) is a multiple of every number between 1 and 13 i.e every number between 1 and 13 is a factor of 13!. If there is a number of the type 13!+a and if "a" is contained in 13! (By that I mean it is one of the integer between 1 and 13,) then "a" will be a factor of 13!+a. In the above question "a" assumes values between 2 and 13, all of which are a part of 13!. Hence the integers between 13!+2 and 13!+13 are not prime. Let me know if you have any doubts.

Regards

Thanks! This helps alot!

gabriel wrote:If there is a number of the type 13!+a and if "a" is contained in 13! (By that I mean it is one of the integer between 1 and 13,) then "a" will be a factor of 13!+a.

Dear Gabriel

Could you pls explain the line above? I am having difficulty understanding this particular problem.