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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 22) or equal to 55 (and the number is divisible by 55). All other digits can be the last digit of a three-digit prime, e.g. 101101, 103103, 107107 and 109109 are prime. Samo's number was odd, so the last digit must have been 55.

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