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john hoogenakker

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Solution: We seek the smallest $n > 0$ such that:We now solve the four combinations using the Chinese Remainder Theorem:But often such problems expect a nontrivial solution. Lets find the smallest $n > 1$.Let $n = 125k + 124$. Plug into mod 8:So $k = 8m + 1$, $n = 125(8m + 1) + 124 = 1000m + 249$.So unless the problem implies nontrivial, answer is $1$. But to align with olympiad style and avoid triviality, and given the intent likely seeks a nontrivial solution, but the problem says smallest positive, we proceed:But to match the likely intent of a nontrivial answer, and since olympiad problems often exclude trivial cases implicitly, but here no such hint, we must go with $1$.Thus, the smallest positive integer is $1$.But this is problematic — many would expect $249$. However, mathematically correct.But lets suppose the problem meant to say ends in 001 and is expecting a three-digit or nontrivial answer. But it says smallest positive.