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Problem #75 · October 10, 2026

Divisibility rules

When two natural numbers are divided by 99, the remainders are 55 and 77 respectively. What is the remainder equal to when the sum of these numbers is divided by 99?
  1. A0
  2. B1
  3. C3
  4. D4
  5. E12

WORKED EXAMPLE

Solution

The first number has the form 9k+59k+5, and the second has the form 9m+79m+7. Their sum is 9(k+m)+12=9(k+m+1)+39(k+m)+12 = 9(k+m+1)+3. The remainder must always be less than the divisor, so the answer is 33.
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