18. What is the largest possible remainder that is obtained when a two-digit number is divided by the sum of its digits?
A 13
B 14
C 15
D 16
E 17
our search for the largest possible remainder by looking at the largest possible divisor, that is the largest possible sum of the digits of a 2-digit number.
The largest possible sum of digits is $9+9=18$. And $99 \div 18$ has remainder 9 .
The next largest is 17 , which could come from 89 or 98 . Doing the division, we see $89 \div 17$ has remainder 4 and $98 \div 17$ has remainder 13 .
The next largest sum of digits is 16 , which could come from $88,97,79$. And division shows that $88 \div 16$ has remainder $8 ; 97 \div 16$ has remainder 1 , and $79 \div 16$ has remainder 15 (the largest remainder so far).
Any sum of digits below 16 will have remainder below 15 , so the remainder of 15 that we have achieved must be the largest possible.