OG11 Question 153

Ok this one is simple if you look at it this way

Lets see what 13! is: 13x12x11 ....x2x1

So it can be written as 13! = 2xQ where Q is an integer
Similarly, it can be written as 3xW where W is an integer

Above listed equations are true because numbers from 2,3,4 to 13 are all factors of 13! so we can write it in the form nR where n is the known factor and R is an integer

K = integer
13! + 2 <= k <= 13! + 13

so if K = 13!+2 = 2xQ+2 = 2(Q+1) not a prime
similarly, if K = 13!+3 = 3W+3 = 3(W+1) not a prime
you can do this upto 13 because you know for sure that all numbers from 2 to 13 are factors of 13!

hence (b) will not be a prime for all given possibilities of K