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Problem 551

AMC 10/12, early questions

Number theory Difficulty 3.5 Multiple choice CEMC Fermat · Canada · 2022

The remainder when 111 is divided by 10 is 1. The remainder when
111 is divided by the positive integer $n$ is 6. The number of possible values of
$n$ is

Pick one

Official solution

Since the remainder when 111 is divided by $n$ is 6, then $111 - 6 = 105$ is a multiple of $n$ and $n>6$ (since, by definition, the
remainder must be less than the divisor).

Since $105 = 3 \cdot 5 \cdot 7$ , the
positive divisors of 105 are $1, 3, 5, 7, 15,
21, 35, 105$.

Therefore, the possible values of $n$
are 7, 15, 21, 35, 105, of which there are 5.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.