Teacher Guide: Divisibility by 4, 6, and 8

Test any whole number for 4, 6 and 8 in seconds: read the last two or three digits, or combine the rules for 2 and 3, and see why it works.

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10 questions on Divisibility. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Test whether a whole number is divisible by 4 using its last two digits, including the tens digit shortcut
Test whether a whole number is divisible by 8 using its last three digits
Test divisibility by 6 by combining the rules for 2 and 3, and explain why both are needed
Explain the rules for 4 and 8 with 100 = 4 × 25 and 1000 = 8 × 125
Apply the rules to leap years and to packing problems with groups of 6 and 8

Prerequisites

• Divisibility Rules: what divisible and remainder mean
• Divisibility by 2, 5, and 10: the last digit rules
• Divisibility by 3 and 9: the digit sum test for 3
• Place value: ones, tens, hundreds, thousands

Discussion Starters

1. Why is the rule for 4 about two digits, the rule for 8 about three digits, but the rule for 2 about only one digit?
2. What would a rule for 16 look like? How many digits would you need to check?
3. Every multiple of 8 is a multiple of 4. Why is the reverse not true?
4. Why can we test 6 with the rules for 2 and 3, but not 8 with the rules for 2 and 4?
5. Someone was born on 29 February 2000. How often do they get a real birthday?

Common Misconceptions

A number ending in 4 or 8 is always divisible by 4 (like 34 or 38)

Remediation: List the multiples of 4 up to 100 and look at their endings: 04, 08, 12, 16, 20... Students see that the last digit alone does not decide, then discover the tens digit pattern (even tens: 0, 4, 8; odd tens: 2, 6).

Divisible by 2 and by 4 means divisible by 8, just like 2 and 3 give 6

Remediation: Test 12, 20 and 28. Then compare factor pairs: 2 and 3 have no common factor, 2 and 4 share 2. Introduce the word coprime.

Every year divisible by 4 is a leap year

Remediation: Show 1900 on an old calendar: February had 28 days. Explain the 400 rule with the average year length 365.2425 days.

Differentiation Ideas

For Struggling Students:

• Start with the multiples of 4 up to 100 on a hundred square and colour them to see the pattern in the last two digits
• Use only four-digit numbers and one rule at a time
• Let students check every verdict with a calculator to build trust in the rules

For On-Level Students:

• Mix tests for 4, 6 and 8 in one set and ask students to name the rule they used
• Solve missing digit problems, like the smallest digit that makes 73_ divisible by 4
• Count the leap years in a century

For Advanced Students:

• Find and prove a rule for 16 (last four digits, because 10,000 = 16 × 625)
• Explain the halving trick for 8 and why three halvings are enough
• Investigate which other divisors can be tested by combining two coprime rules

Standards Alignment

4.OA.B.4 (CCSS.MATH.CONTENT.4.OA.B.4)
Find all factor pairs for a whole number. Recognize that a whole number is a multiple of each of its factors.
4.NBT.B.6 (CCSS.MATH.CONTENT.4.NBT.B.6)
Find whole-number quotients and remainders using strategies based on place value.
6.NS.B.4 (CCSS.MATH.CONTENT.6.NS.B.4)
Find the greatest common factor and least common multiple of two whole numbers and use the distributive property with common factors.

Lesson Resources

visualDivisibility explainer
Shows 45,136 = 45 × 1000 + 136 and highlights the digits that each rule for 2, 4 and 8 reads
gameGuess 2, 3 and 6
Students guess yes or no for 2, 3 and 6 on each number, then reveal the rule and score
activityLeap year hunt
Students list the birth years of family members and test which ones were leap years
worksheetPacking problems
Orders of eggs, muffins and bottles to pack in boxes of 6 and crates of 8

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A divisibility rule tells you whether a number is divisible by another number (division with remainder

0

) without doing the whole division. In this lesson you learn the rules for

4

8

and

6

Divisible by 4: the last two digits

A whole number is divisible by

4

when the number formed by its last two digits is divisible by

4

7316: \quad 16 = 4 \times 4 \;\Rightarrow\; 7316 \div 4 = 1829

Why?

7316 = 73 \times 100 + 16

, and

100 = 4 \times 25

. Every whole number of hundreds is a multiple of

4

, so only the last two digits can leave a remainder.

Quick check with the tens digit: if the tens digit is even, the last digit must be

0

4

8

. If the tens digit is odd, the last digit must be

2

6

Divisible by 8: the last three digits

A whole number is divisible by

8

when the number formed by its last three digits is divisible by

8

, because

1000 = 8 \times 125

45{,}136: \quad 136 = 8 \times 17 \;\Rightarrow\; 45{,}136 \div 8 = 5642

Divisible by 6: the rules for 2 and 3 together

A whole number is divisible by

6

when it is divisible by

2

and by

3

: the last digit is even and the digit sum is divisible by

3

. This works because

6 = 2 \times 3

and

2

and

3

are coprime.

5742: \quad \text{last digit } 2 \text{ (even)}, \quad 5 + 7 + 4 + 2 = 18 = 3 \times 6 \;\Rightarrow\; 5742 \div 6 = 957

A chain of rules

\text{divisible by } 8 \;\Rightarrow\; \text{divisible by } 4 \;\Rightarrow\; \text{divisible by } 2

The chain only goes one way:

12

is divisible by

4

but not by

8

Worked Examples

7316

divisible by

4

Split off the last two digits

7316 = 73 \times 100 + 16

→ Last two digits:

16

Remember why the hundreds do not matter

100 = 4 \times 25

, so

7300

is a multiple of

4

→ Only

16

decides

Test the last two digits

16 = 4 \times 4

→

16

is divisible by

4

Conclude and check

7316 \div 4 = 1829

→ Divisible, remainder

0

Answer: Yes. The last two digits form 16, which is divisible by 4, so 7316 is divisible by 4 (7316 ÷ 4 = 1829).

Common Mistakes

Checking only the last digit for $4$ : " $2134$ ends in $4$ , so it is divisible by $4$ "

Why it's wrong: The rule for $4$ reads the last two digits, because $10$ is not a multiple of $4$ . Here $34 = 4 \times 8 + 2$ .

Correct: Test the last two digits: $34$ leaves remainder $2$ , so $2134$ is not divisible by $4$ .

Checking only the last two digits for $8$ : " $3124$ ends in $24 = 8 \times 3$ , so it is divisible by $8$ "

Why it's wrong: $100$ is not a multiple of $8$ ( $100 = 8 \times 12 + 4$ ), so the hundreds digit still matters. The rule for $8$ needs the last three digits.

Correct: Test $124$ : $124 = 8 \times 15 + 4$ . So $3124$ is not divisible by $8$ (remainder $4$ ).

Thinking that divisible by $2$ and by $4$ means divisible by $8$

Why it's wrong: $12$ is divisible by $2$ and by $4$ , but $12 \div 8 = 1.5$ . Combining rules only works when the two numbers are coprime, like $2$ and $3$ for $6$ . $2$ and $4$ share the factor $2$ .

Correct: For $8$ use its own rule (last three digits). For $6$ you may combine $2$ and $3$ .

Testing only one part of the rule for $6$ : " $27$ is divisible by $3$ , so it is divisible by $6$ "

Why it's wrong: $6$ needs both tests. $27$ is odd, so it fails the test for $2$ . Likewise $26$ is even but its digit sum $8$ fails the test for $3$ .

Correct: A number is divisible by $6$ only if it is even and its digit sum is divisible by $3$ , for example $5742$ .

Why It Matters

These three rules show up whenever things come in groups of 4, 6 or 8:

Calendars: a year is a leap year when it is divisible by $4$ , with a special rule for century years such as $1900$ and $2000$
Packing: eggs and drinks are sold in packs of $6$ , crayons and bottles in boxes or crates of $8$ . Can $1734$ muffins fill boxes of $6$ exactly?
Computers: $1$ byte is $8$ bits, and memory sizes are multiples of $8$
Fractions: spotting a factor $4$ , $6$ or $8$ in a numerator and denominator lets you simplify quickly, like $\frac{24}{32} = \frac{3}{4}$ after dividing both by $8$
Checking work: a quick look at the last digits tells you if an answer like $7316 \div 4$ can come out whole before you calculate it

The ideas behind the rules (

100 = 4 \times 25

1000 = 8 \times 125

6 = 2 \times 3

) are also the first step toward the rules for 12, 15, 18 and beyond.

Real World Applications

Leap years in a century

The calendar adds a day (29 February) in every year divisible by 4, but skips century years that are not divisible by 400.

Example:

$2024$ is a leap year ( $24 = 4 \times 6$ ), $2000$ was one ( $2000 = 400 \times 5$ ), but $1900$ was not ( $1900 \div 400 = 4.75$ ).

1Try It Yourself

You count the leap years from 2001 to 2100, including 2100.

How many leap years are there in this period?

Step 1: Write the mathematical expression

Type the number of leap years:

Packing in sixes

Bakeries, egg farms and drink makers pack in 6s. The rules for 2 and 3 tell them before counting whether nothing will be left over.

Example:

$4320$ eggs fill boxes of $6$ exactly: $4320$ is even and $4 + 3 + 2 + 0 = 9$ is divisible by $3$ , so $4320 \div 6 = 720$ boxes.

2Try It Yourself

A bakery has 1734 muffins and packs them in boxes of 6.

How many full boxes can it fill, with no muffins left over?

Step 1: Write the mathematical expression

Write the division that gives the number of boxes:

Crates of eight

Bottles and cans often travel in crates of 8. Only the last three digits of a large order decide whether the crates come out even.

Example:

$1560$ pencils fit in boxes of $8$ : the last three digits $560 = 8 \times 70$ , so $1560 \div 8 = 195$ boxes.

3Try It Yourself

A warehouse receives 23,512 bottles to load into crates of 8.

How many full crates does it fill?

Step 1: Write the mathematical expression

Write the division that gives the number of crates:

Key Takeaways

1Divisible by $4$ : the last two digits form a number divisible by $4$ , because $100 = 4 \times 25$
2Tens digit shortcut for $4$ : even tens digit needs last digit $0$ , $4$ or $8$ ; odd tens digit needs $2$ or $6$
3Divisible by $8$ : the last three digits form a number divisible by $8$ , because $1000 = 8 \times 125$
4Divisible by $6$ : even and digit sum divisible by $3$ (both tests must pass)
5Divisible by $8$ implies divisible by $4$ , which implies divisible by $2$ , but not the other way round ( $12$ )
6Leap years: divisible by $4$ , but century years only when divisible by $400$ ( $2000$ yes, $1900$ no)

Frequently Asked Questions

Why do the last two digits decide divisibility by 4?

Any whole number is a number of hundreds plus its last two digits:

7316 = 73 \times 100 + 16

. Since

100 = 4 \times 25

, every whole number of hundreds is divisible by

4

. The remainder of the whole number divided by

4

is therefore the same as the remainder of the last two digits. The same idea with

1000 = 8 \times 125

gives the rule for

8

If a number is divisible by 4, is it also divisible by 8?

Not always.

12

20

and

28

are divisible by

4

but not by

8

. The chain only goes downward: every multiple of

8

is a multiple of

4

(because

8 = 4 \times 2

), and every multiple of

4

is even.

Can I build other rules the way 6 = 2 and 3 works?

Yes, but only with coprime numbers (no common factor other than

1

6 = 2 \times 3

works because

2

and

3

are coprime.

8 = 2 \times 4

does not work, because

2

and

4

share the factor

2

12

passes both tests and is still not divisible by

8

. The lesson Combined Divisibility Rules shows this for

12

15

18

and

36

Why must century years be divisible by 400 to be leap years?

A year lasts about

365.2422

days. Adding a leap day every

4

years gives an average of

365.25

days, a bit too long. Skipping

3

leap days every

400

years (

1700

1800

1900

, but not

2000

) leaves

97

leap years in

400

years:

365 + \frac{97}{400} = 365.2425

days, very close to the real value.

Glossary

Divisibility rule: A shortcut that tells whether a number is divisible by another without doing the full division, for example the last two digits for $4$ .
Divisible by: A whole number is divisible by $n$ when dividing it by $n$ leaves remainder $0$ : $7316$ is divisible by $4$ .
Remainder: What is left after dividing as many whole times as possible: $34 \div 4 = 8$ remainder $2$ .
Digit sum: The sum of all digits of a number: the digit sum of $5742$ is $5 + 7 + 4 + 2 = 18$ .
Coprime: Two whole numbers are coprime when their only common factor is $1$ , like $2$ and $3$ . Then divisible by both means divisible by their product.
Leap year: A year with $366$ days (29 February added): divisible by $4$ , but a century year only when divisible by $400$ .

Formula Card

Divisible by 4

Last two digits divisible by

4

Test the last two digits, because $100 = 4 \times 25$

Divisible by 8

Last three digits divisible by

8

Test the last three digits, because $1000 = 8 \times 125$

Divisible by 6

Divisible by

2

and by

3

Last digit even and digit sum divisible by 3

Leap year

Divisible by

4

; century years: by

400

2024 and 2000 are leap years, 1900 and 2100 are not

Teacher Guide: Divisibility by 4, 6, and 8

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Divisible by 4: the last two digits

Divisible by 8: the last three digits

Divisible by 6: the rules for 2 and 3 together

A chain of rules

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Leap years in a century

Packing in sixes

Crates of eight

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Divisible by 4

Divisible by 8

Divisible by 6

Leap year

More in This Topic

Divisibility Rules

Divisibility by 2, 5, and 10

Divisibility by 3 and 9

Combined Divisibility Rules