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South African Mathematics Olympiad Second Round

Number theory Difficulty 4.5 AIME Prove it South Africa

Let $P$ be the product of any three consecutive positive odd integers. What is the highest common factor (greatest common divisor) of all such numbers $P$ ?

Solution

For any three consecutive odd integers, exactly one is divisible by $3$ , so $3$ is a common factor of all such numbers $P$ . Even the first two non-overlapping values $P = 1 \times 3 \times 5$ and $P = 7 \times 9 \times 11$ have highest common factor $3$ , which is therefore the highest common factor of all values of $P$ .

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