﻿ Divisibility

# 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 $\,3\,.$

☆ (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!

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 $\,5\,.$
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 . 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.