# Divisibility

The numbers
$0$, $2$, $4$, $6$, … are called
*even numbers*.
Even numbers always end in one of these digits:
$0$, $2$, $4$, $6$, or
$\,8\,.$

Even numbers can always be divided into two equal (even) piles.

Note:
The numbers
$1$, $3$, $5$, $7$, … are called *odd numbers*.

The numbers
$0$, $2$, $4$, $6$, … are also said to be
*divisible by 2*.
*Divisible by 2* means that
$\,2\,$ goes into the number evenly.
The phrases *even* and *divisible by 2* are interchangeable.

*Divisible by 3* means that
$\,3\,$ goes into the number evenly.
*Divisible by 4* means that
$\,4\,$ goes into the number evenly; and so on.

A *divisibility test* is a shortcut to decide if a number is divisible by a given number.

Also, if a number is divisible by $\,2\,,$ then it ends in $0$, $2$, $4$, $6$, or $\,8\,.$

For example,
$\,87{,}356\,$ is divisible by $\,2\,,$ since it ends in the digit $\,6\,.$
However,
$\,87{,}357\,$ is *not* divisible by $\,2\,,$ since it ends in the digit $\,7\,.$

There's a neat trick for deciding if a number is divisible by $\,3\,.$ The technique is illustrated with the following example:

## Example

*without*using your calculator!)

If this final number is divisible by $\,3\,,$ then the number you started with is also divisible by $\,3\,.$

If this final number is not divisible by $\,3\,,$ then the number you started with is not divisible by $\,3\,.$

☆ (Speed-it-up trick!)

Let's re-do the previous example, being a bit more clever.
You don't really have to add up *all* the digits!

Looking at the number $\,57{,}394\,,$ the digits $\,3\,$ and $\,9\,$ are clearly divisible by $\,3\,.$
So, don't bother including them in your sum!

That leaves you with $\,5\,,$ $\,7\,$ and $\,4\,.$
But, the sum of $\,5\,$ and $\,7\,$ is $\,12\,,$ which is divisible by $\,3\,.$
So, you're *really* only left with the digit $\,4\,,$ which is clearly not divisible by $\,3\,.$

If you get into the habit of discarding $\,3\,$’s, $\,6\,$’s, $\,9\,$’s,
and obvious sums that give a multiple of $\,3\,$ (like $\,5 + 7\,$),
then this test can go much faster!

Read the text for a proof of the "divisibility by 3" test (on page 14).

Also, if a number is divisible by $\,5\,,$ then it ends in $\,0\,$ or $\,5\,.$

Also, if a number is divisible by $\,10\,,$ then it ends in $\,0\,.$

More compactly,
we can say that *
a number is divisible by 10
if and only if it ends with a 0
*.
The phrase ‘ if and only if ’ will be thoroughly discussed in
a future section.

As an aside, you can read the text for a clever ‘finger trick’ for multiplying by 9 (on page 15).

## Practice

Decide if the number is divisible by: 2, 3, 5, 10. Check all appropriate boxes.