B. Number · Operations: Divisibility, Decimals, Fractions, Ratios and Percents

Divisibility Rules for 2, 4, 5, 8 and 10

B2.2About 40 minutes

Learning goals

I can explain what it means for one number to be divisible by another.
I can use rules to tell quickly if a number is divisible by 2, 5 or 10.
I can use rules to tell if a number is divisible by 4 or 8, and explain why the rules work.

Words to know

divisible
One whole number is divisible by another if it divides with no remainder. 24 is divisible by 4 because 24 ÷ 4 = 6.

divisibility rule
A shortcut that tells you if a number is divisible by another without doing the whole division.

factor
A number that divides another number with no remainder. 4 is a factor of 24.

multiple
The product of a number and a whole number. 24 is a multiple of 4.

even number
A whole number that is divisible by 2. It ends in 0, 2, 4, 6 or 8.

Let's start with a gym class

132 students are coming to a school sports day. The coach wants teams of 4 with nobody left out. Will it work?

You could do the division. But there is a faster way. Look at just the last two digits: 32. 32 ÷ 4 = 8, with nothing left over. So 132 is divisible by 4, and the teams will work out (there will be 33 teams).

Shortcuts like this are called divisibility rules. In this lesson you will learn the rules for 2, 4, 5, 8 and 10, and why they work.

What does divisible mean?

A number is divisible by another number if you can divide it into equal groups with nothing left over.

35 ÷ 5 = 7. No remainder, so 35 is divisible by 5.
37 ÷ 5 = 7 R2. There is a remainder, so 37 is not divisible by 5.

If 35 is divisible by 5, then 5 is a factor of 35, and 35 is a multiple of 5.

Rules for 2, 5 and 10: look at the last digit

2: the last digit is 0, 2, 4, 6 or 8 (an even number).
5: the last digit is 0 or 5.
10: the last digit is 0.

Why? Any number of tens can always be split into groups of 2, of 5 or of 10, because 10 = 2 × 5. So the tens, hundreds and thousands never leave anything over. Only the ones digit can leave a remainder. That's why you only need to look at the last digit.

Split 3518 into 3510 + 8. The tens part always divides evenly by 2, 5 and 10, so the ones digit decides.

Rules for 4 and 8: look at the last two or three digits

4: the number made by the last two digits is divisible by 4.
8: the number made by the last three digits is divisible by 8.

Why? 100 = 4 × 25, so every hundred splits into groups of 4 with nothing left over. Only the last two digits can leave a remainder.

1000 = 8 × 125, so every thousand splits into groups of 8. Only the last three digits can leave a remainder.

Bars drawn to scale. Hundreds always split into 4s and thousands always split into 8s, so only the small part on the right needs checking: 32 for 4, and 344 for 8.

Divisible by	Look at	Rule	Yes example	No example
2	last digit	0, 2, 4, 6 or 8	3 518	3 517
5	last digit	0 or 5	4 265	4 262
10	last digit	0	7 090	7 095
4	last two digits	divisible by 4	2 716 (16 ÷ 4 = 4)	2 714 (14 ÷ 4 = 3 R2)
8	last three digits	divisible by 8	5 240 (240 ÷ 8 = 30)	5 220 (220 ÷ 8 = 27 R4)

Divisibility rules for 2, 4, 5, 8 and 10.

Worked example

Example 1: Divisible by 4?

Is 3 516 divisible by 4?

Look at the last two digits: 16.
16 ÷ 4 = 4, with no remainder.
Check with a calculator if you like: 3 516 ÷ 4 = 879.

Yes, 3 516 is divisible by 4.

Worked example

Example 2: Divisible by 8?

Is 7 344 divisible by 8?

Look at the last three digits: 344.
Split 344 into friendly parts: 320 + 24. 320 ÷ 8 = 40 and 24 ÷ 8 = 3. So 344 ÷ 8 = 43, with no remainder.

Yes, 7 344 is divisible by 8. (7 344 ÷ 8 = 918.)

Worked example

Example 3: Test one number for several rules

Is 2 630 divisible by 2, 5, 10 and 4?

Last digit is 0: divisible by 2, by 5 and by 10.
Last two digits are 30. 30 ÷ 4 = 7 R2. Not divisible by 4.

2 630 is divisible by 2, 5 and 10, but not by 4.

Watch out

Being even is not enough for 4 or 8. 30 is even, but 30 is not divisible by 4. Always check the last two digits for 4 and the last three digits for 8.

Just for fun

132 students make 33 perfect teams of 4. Teams of 5? That would be 26 teams, with 2 students left over to be the official cheerleaders.

Hands-on activity

Divisibility detective

You need: a deck of cards with the 10s and face cards removed (aces count as 1), a scrap of paper with 0 written on it to use as a zero card (shuffle it into the deck), paper, pencil and a calculator for checking.

Deal 4 cards to make a 4-digit number. Write it down.
Use the rules to predict: is it divisible by 2? 4? 5? 8? 10?
Check each prediction with the calculator. Did you get a whole number answer?
Now try to rearrange the same 4 cards to make a number that is divisible by 4. Then try for 8. (Sometimes it is not possible. Can you explain why?)

Math talk: "Which rule was the easiest? Which one took the most thinking?"

Your turn

Is 785 divisible by 5?

Is 4 312 divisible by 2?

Is 516 divisible by 4?

Is 1 250 divisible by 10?

Is 738 divisible by 4?

Is 5 120 divisible by 8?

Is 9 432 divisible by 8?

More practice: warm-up

Is 46 divisible by 2?

Is 95 divisible by 10?

Is 340 divisible by 5?

Is 77 divisible by 2?

Is 1 000 divisible by 10?

More practice: on your own

Is 2 724 divisible by 4?

Is 6 150 divisible by 4?

Is 3 096 divisible by 8?

Is 14 340 divisible by 8?

Which number is divisible by both 2 and 5: 315, 320 or 324?

Challenge

Which digits can go in the blank so that 47_ is divisible by 4?

Explain why every number that is divisible by 8 is also divisible by 4.

336 students are going on a trip. Can they be split evenly into groups of 2? 4? 5? 8? 10?

Check your understanding

Quiz

Which number is divisible by 4?

Which number is divisible by 5 but not by 10?

To test if a number is divisible by 8, which digits do you look at?

Is 6 128 divisible by 8?

Why does the rule for 4 use only the last two digits?

Which number is divisible by 2, 5 and 10?

For parents and teachers

Big idea: The Ontario curriculum (B2.2) asks Grade 6 students to understand the divisibility rules and use them to determine whether numbers are divisible by 2, 3, 4, 5, 6, 8, 9 and 10. Understand means being able to say why a rule works, not only memorize it. This lesson covers 2, 4, 5, 8 and 10. The next lesson covers 3, 6 and 9.

Common difficulties:

Thinking any even number is divisible by 4. 30 and 50 are even but not divisible by 4.
Looking at the last digit only for 4 and 8. 18 ends in 8 but is not divisible by 8.
Forgetting a 0 in the last three digits: in 3 096, the last three digits are 096 = 96.

How to help: Ask divisibility questions in daily life: "Can 28 cookies be shared equally among 4 people? Among 8?" Let your child predict with a rule, then check with a calculator. Ask "Why does the rule work?" and listen for "because 10 (or 100, or 1000) splits evenly".