South African Mathematics Olympiad Second Round

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 $P$.