Show that $(a − b)(b − c)(a − c)$ is always an even integer if $a, b,$ and $c$ are integers.

Question

Show that $(a − b)(a − c)(b − c)$ is always even if $a, b, c$ are integers. It should be solved with the pigeonhole principle but I could not find a solution.

Answer 1

If a, b and c are odd or even it is obvius that difference between each pair is even number.

If there are two odd numbers and an even number, the difference between the odd numbers is even.

If there are two even numbers and an odd number, the difference between the even numbers is even.

Answer 2

Hint: Remember that when adding/multiplying integers,

• $\text{even} \pm \text{even} = \text{even}$
• $\text{odd} \pm \text{odd} = \text{even}$
• $\text{even} \pm \text{odd} = \text{odd}$
• $\text{even} \times \text{even} = \text{even}$
• $\text{even} \times \text{odd} = \text{even}$
• $\text{odd} \times \text{odd} = \text{odd}$