What is the largest possible remainder when a two-digit number is divided by the sum of its digits?

a) 13

b) 14

c) 15

d) 16

e) 17

The largest possible digit sum = 9 + 9 = 18, so no remainder can be > 17.

Working down from the top:

99/18 → remainder 9

98/17 → remainder 13

89/17 → remainder 4

97/16 → remainder 1

88/16 → remainder 8

79/16 → remainder 15

After that the divisor is 15 or less,

so the remainder is 14 or less,

and so the largest remainder is 15,

from 79 = 4 * 16 + 15.