Competition problem — Number theory

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Number theory Difficulty 4.0 AMC 10/12 Find the answer South Africa

If $\frac{n+3}{13}$ is a positive integer, then the remainder when $n$ is divided by $13$ is

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

10 Suppose $\frac{n+3}{13} = k$ ; then $n = 13k - 3$ or $n = 13(k-1) + 10$ . This shows that $n$ leaves remainder $10$ on division by $13$ .

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