Ok...so to answer this question, let's first state some rules....
3 divides any one number in a set of three consecutive integers so;
3 either divides 1,2 or 3 or 2,3,4 or 3,4,5 (you get the idea)
so 3 divides either n-1,n or n+1
Condition 1: 3 does not divide n^2 + n or n(n+1) so basically it implies that 3 does not divide n or n+1, so 3 MUST divide n-1. The only exceptions are -1,0,1 since 3 does divide 0*1 but does not divide -1 and the other exception is -2,-1,0 since 3 divides -1*0 but 3 does not divide -2. These two exceptions contradict the case for all other cases like 1,2,3 or -1,-2,-3 etc....so this alone does not help us. We need to satisfy some other conditions that eliminates zero as the third integer.
Condition 2: 3n + 5 > k + 8, k is a multiple of 3 so let k = 3c, where c = set of all positive real numbers
so 3n + 5 > 3c + 8
3n > 3c + 3
n > c + 1.
so we know that if k = 3 is the smallest possible multiple of 3 so c = 1. so n>2 which means that we can now positively say that both statements together state that 3 divides n-1.
Hence the answer is C.
