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.

DIVISIBILITY BY 2

If a number ends in

$0$, $2$, $4$, $6$, or $8$, then the number is divisible by

$\,2\,$.
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:

Question: Is $\,57{,}394\,$ divisible by $\,3\,$?
(Answer without using your calculator!)

Solution: Add up the digits in the number:

$5 + 7 + 3 + 9 + 4 = 28\;$.
The sum is $\,28\,$; add up the digits again:

$2 + 8 = 10\,$.
Since

$\,10\,$ is not divisible by $\,3\,$, the original number

$\,57{,}394\,$ is also not divisible by $\,3\,$.

DIVISIBILITY BY 3

To decide if a number is divisible by

$\,3\,$, add up the digits in the number.
Continue this process of adding the digits until you get a manageable number.
(If you want, keep going until you get a single-digit number.)
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 $\,3\,$.

Read the text for a proof of the "divisibility by 3" test.

DIVISIBILITY BY 5

If a number ends in

$\,0\,$ or $\,5\,$, then the number is divisible by $\,5\,$.
Also, if a number is divisible by

$\,5\,$, then it ends in $\,0\,$ or $\,5\,$.

Read the text for a clever "finger trick" for divisibility by 9.

DIVISIBILITY BY 10

If a number ends in $\,0\,$, then the number is divisible by $\,10\,$.
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.

Master the ideas from this section
by practicing both exercises at the bottom of this page.

When you're done practicing, move on to:
Basic Properties of Zero and One

(MAX is 12; there are 12 different problem types.)