Competition problem — Number theory

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Number theory Difficulty 3.6 AMC 10/12 Find the answer Philippines

Problem:

Let $n$ be a positive integer greater than $1$ . If $2n$ is divided by $3$ , the remainder is $2$ . If $3n$ is divided by $4$ , the remainder is $3$ . If $4n$ is divided by $5$ , the remainder is $4$ . If $5n$ is divided by $6$ , the remainder is $5$ . What is the least possible value of $n$ ?

Your answer

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

Solution

Solution:

We have $2n \equiv 2 \pmod{3}$ , $3n \equiv 3 \pmod{4}$ , $4n \equiv 4 \pmod{5}$ , and $5n \equiv 5 \pmod{6}$ . This is equivalent to saying that $n \equiv 1 \pmod{3,4,5,6}$ . The smallest such $n$ is one more than the $\mathrm{LCM}$ of $3,4,5,$ and $6$ , which is $61$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.