Question 2

An integer $$M$$ is divisible by 4 but not by 256. What is the number of distinct possible remainders when $$M$$ is divided by 256?

Correct Answer: 63

Solution

Write $$M = 256q + r$$ with $$0 \le r < 256$$. Both $$M$$ and $$256q$$ are divisible by 4, so $$r$$ is divisible by 4, and $$r \ne 0$$ because $$M$$ is not divisible by 256. The possible remainders are therefore $$4, 8, 12, \ldots, 252$$, and each one occurs, giving $$252/4 = 63$$ values.

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