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National Math Olympiad in Slovenia

Number theory Difficulty 3.8 AMC 10/12 Find the answer Slovenia

Samo wrote down a 3-digit odd positive integer on a piece of paper. He then told Peter what the last digit of this number was. Peter immediately concluded that the number Samo wrote down is not prime. What was this last digit?

Pick one

Solution

Peter can only arrive at this conclusion if the last digit is even (and the number is divisible by $2$ ) or equal to $5$ (and the number is divisible by $5$ ). All other digits can be the last digit of a three-digit prime, e.g. $101$ , $103$ , $107$ and $109$ are prime. Samo's number was odd, so the last digit must have been $5$ .

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