A two digit number is divided by the sum of digits. What is the maximum possible remainder ?

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Solution

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.

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