Competition problem — Number theory

Library / /16 of 20

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 $P$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.