24. When 5655 is divided by a two-digit positive integer $N$, the remainder is 11 . When 5879 is divided by the same positive integer $N$, the remainder is 14 .
What is the sum of the digits of $N$ ?
A 6
B 7
C 8
D 9
E 10
24. C We are given that $5655=N \times j+11$ and $5879=N \times k+14$.
Subtracting the first equation from the second, we obtain, $224=N \times(k-j)+3$. Therefore $221=N \times(k-j)$.
It follows that $N$ is a positive factor of 221 , so that $N$ is equal to $1,13,17$ or 221 . But one of the remainders is 14 , so $N$ is equal to 17 or 221 . However, 221 does not have two digits.
Therefore $N=17$ and the sum of its digits is $1+7=8$.