Competition problem — Number theory

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Number theory Difficulty 3.9 AMC 10/12 Find the answer North Macedonia

Which digit has to be put in the place of the star in the number $2008*$ in order to be divisible with $6$ ?

Your answer

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

Solution

In order $2008*$ to be divisible with $6$ it has to be divisible with $2$ and $3$ . In order to be divisible with $2$ it has to end on $0$ , $2$ , $4$ , $6$ or $8$ . From the numbers $20080$ , $20082$ , $20084$ , $20086$ and $20088$ only $20082$ and $20088$ are divisible with $3$ . So on the place of the $*$ we can put $2$ or $8$ .

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