Competition problem — Number theory

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Number theory Difficulty 4.7 AIME Prove it Belarus

Prove that for any positive integer $n$ there exist coprime positive integers numbers $a \neq b$ such that for each $k$ from $1$ to $n$ the numbers $a+k$ and $b+k$ are not coprime.

Solution

Let $a = 1$ and $b = 1 + (n + 1)$ , obviously they are coprime. Moreover, for any $k$ from $1$ to $n$ both numbers $a + k$ and $b + k$ are divisible by $k + 1$ and are not coprime.

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