Question

Prove that the product of any three consecutive numbers is even.

Answer 1

Answer:

The product of three consecutive numbers is essentially the product of an odd number and an even number.

The product of an odd number and an even number is always even.

Explanation:

Consecutive numbers follow a pattern - they alternate between odd and even. (Any number sequence goes: ... odd, even, odd, even, odd ...)

With three consecutive numbers, there are two combinations:

1. odd, even, odd

2. even, odd, even

Remember these rules:

The product of two odd numbers is always odd.

The product of two even numbers is always even.

***The product of an odd number and an even number is always even.

Simplify the equations of consecutive numbers with the rules.

odd X even X odd

= even X (odd)(odd)

= even X odd

= even

even X odd X even

= odd X (even)(even)

= odd X even

= even

***Why this rule works:

In any multiplication statement, the multiplication sign means "groups of".

For example, 5 times 3 = 5 groups of 3.

An even number means, like if a number was represented by number of people, everyone would have a partner.

odd X even means "odd# groups of even # people".

Within each group, everyone already has a partner. It does not matter how many groups there are.  Therefore it's an even number.

even X odd means "even# groups of odd # people".

In this model, there is a leftover person in each of the groups. These people find a partner from another group. Since there are an even number of groups, there is an even number of leftover people. Then they all find a partner. Therefore it's an even number.