12. For how many positive integers $N$ is the remainder 6 when 111 is divided by $N$ ?
A 5
B 4
C 3
D 2
E 1
Solution
A

The remainder when 111 is divided by $N$ is 6 provided that $111=Q N+6$, where $Q$ is a non-negative integer and $6<N$. In other words, $N$ is a factor of $111-6$ with $6<N$.
Now $111-6=105$. The prime factorization of 105 is $3 \times 5 \times 7$. Therefore the factors of $111-6$ are $1,3,5,7,15,21,35$ and 105 .
Of these 8 factors all but 1,3 and 5 are greater than 6 .
Therefore there are 5 positive integers $N$ which give a remainder 6 when 111 is divided by $N$.