Teacher Guide: Divisibility Rules

Learn quick tricks to tell if a number divides evenly into another without doing long division.

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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

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

For Teachers

Learning Objectives

Apply divisibility rules for 2, 3, 4, 5, 6, 9, and 10
Determine whether a number is divisible by a given factor
Use divisibility rules to solve real-world problems
Explain why divisibility rules work for sums of digits

Prerequisites

• Understanding of division with remainders
• Knowledge of even and odd numbers
• Ability to add multi-digit numbers

Discussion Starters

1. Which divisibility rule do you find easiest to use? Why?
2. Can you think of a situation where knowing if a number is divisible by 5 would be helpful?
3. Why do you think there's no simple 'last digit' rule for divisibility by 3?
4. If a number is divisible by both 4 and 3, is it divisible by 12? Can you find an example?

Common Misconceptions

A number ending in 3 is divisible by 3

Remediation: Show counterexamples like 13 and 23. Then show that 12 (not ending in 3) IS divisible by 3. Emphasize: add ALL digits!

If divisible by 2 and 4, then divisible by 8

Remediation: Use 12 as a counterexample: 12 is divisible by 2 and 4, but not by 8 (12/8 = 1 R 4). Rules don't always combine!

Differentiation Ideas

For Struggling Students:

• Start with only divisibility by 2 and 5 (last digit rules)
• Use color-coded digit cards for adding digits
• Practice with two-digit numbers before moving to larger ones

For On-Level Students:

• Apply all divisibility rules to three and four-digit numbers
• Determine all factors of a number using divisibility rules
• Solve word problems involving splitting items into groups

For Advanced Students:

• Explore divisibility rules for 7, 11, and 12
• Investigate why the rules work using place value
• Create divisibility puzzles for classmates

Standards Alignment

4.OA.B.4 (CCSS.MATH.CONTENT.4.OA.B.4)
Find all factor pairs for a whole number in the range 1-100
6.NS.B.2 (CCSS.MATH.CONTENT.6.NS.B.2)
Fluently divide multi-digit numbers using the standard algorithm

Lesson Resources

visualInteractive Factor Tree
Students build factor trees to find prime factors
activityDivisibility Sorting Game
Sort numbers by which rules they follow
worksheetDivisibility Practice
Apply all divisibility rules to multi-digit numbers

Lesson Content

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

Definition

A divisibility rule is a shortcut that tells you whether one number divides evenly into another without leaving a remainder.

When a number divides evenly, we say the larger number is divisible by the smaller one.

36 \div 4 = 9 \text{ (no remainder)} \Rightarrow 36 \text{ is divisible by } 4

37 \div 4 = 9 \text{ R } 1 \Rightarrow 37 \text{ is NOT divisible by } 4

Worked Examples

Is $246$ divisible by $2$ ?

Look at the last digit

The last digit of $246$ is $6$ → Last digit: $6$

Check if it's even (0, 2, 4, 6, or 8)

$6$ is an even number → Yes, $6$ is even

Apply the rule

If the last digit is even, the whole number is divisible by $2$

Answer: Yes, $246$ is divisible by $2$ because it ends in $6$ (an even digit).

Common Mistakes

Using the last digit rule for divisibility by 3

Why it's wrong: Unlike divisibility by 2 or 5, the last digit alone doesn't tell you about divisibility by 3. For example, $13$ ends in $3$ but is NOT divisible by $3$ .

Correct: For divisibility by 3, add ALL the digits together and check if that sum is divisible by 3.

Checking only divisibility by 2 OR 3 for divisibility by 6

Why it's wrong: A number must pass BOTH tests. For example, $9$ is divisible by $3$ but not by $2$ , so it's not divisible by $6$ .

Correct: For divisibility by 6, the number must be divisible by BOTH 2 AND 3.

Confusing the rules for 3 and 9

Why it's wrong: Both rules involve summing digits, but they check different divisors. A number divisible by 9 is always divisible by 3, but not vice versa.

Correct: For 3: digit sum divisible by 3. For 9: digit sum divisible by 9. Example: $12$ (digits sum to $3$ ) is divisible by $3$ but not by $9$ .

Looking at only the last digit for divisibility by 4

Why it's wrong: Unlike 2 and 5, you need the last TWO digits for 4. For example, $14$ ends in $4$ but $14 \div 4 = 3$ R $2$ .

Correct: For divisibility by 4, check if the last TWO digits form a number divisible by 4.

Why It Matters

Divisibility rules save time and mental energy:

Simplifying fractions: Quickly find common factors to reduce $\frac{12}{18}$ to $\frac{2}{3}$
Mental math: Know instantly that 150 people can be split into groups of 5
Prime factorization: Find factors faster when building factor trees
Real life: Split a bill of 45 dollars among 5 friends? Yes, it works out evenly!

These rules are like mental shortcuts that mathematicians use every day.

Real World Applications

Splitting Costs Evenly

When splitting a bill among friends, divisibility rules help you know instantly if it will work out evenly.

Example:

A restaurant bill is 72 dollars. Can 6 people split it evenly? Check: $72$ ends in $2$ (divisible by $2$ ) and $7 + 2 = 9$ (divisible by $3$ ), so yes, divisible by $6$ . Each person pays 12 dollars.

1Try It Yourself

A group of 9 friends wants to split a bill of 108 dollars evenly.

Can they do it? How much does each person pay?

Step 1: Write the mathematical expression

First check divisibility by 9:

Organizing Items in Groups

Teachers, coaches, and event planners use divisibility to organize people or items into equal groups.

Example:

A teacher has 28 students. Can she make groups of 4? Check: Last two digits are $28$ , and $28 \div 4 = 7$ . Yes! She can make 7 groups of 4.

2Try It Yourself

A coach has 45 players and wants to form teams of 5.

Is this possible? How many teams can be formed?

Step 1: Write the mathematical expression

Check divisibility by 5:

Key Takeaways

1Divisibility rules are shortcuts to check if one number divides evenly into another
2Divisible by 2: last digit is even (0, 2, 4, 6, 8)
3Divisible by 3: sum of all digits is divisible by 3
4Divisible by 4: last two digits form a number divisible by 4
5Divisible by 5: last digit is 0 or 5
6Divisible by 6: divisible by both 2 AND 3
7Divisible by 9: sum of all digits is divisible by 9
8Divisible by 10: last digit is 0

Frequently Asked Questions

Why does the sum of digits work for 3 and 9?

It's related to place value! Each place (ones, tens, hundreds) leaves a remainder of 1 when divided by 9 (or 3). So the remainder of the whole number equals the remainder of the digit sum.

Is there a rule for divisibility by 7?

Yes, but it's more complicated! Double the last digit, subtract it from the rest, and check if the result is divisible by 7. For most cases, it's easier to just divide.

If a number is divisible by 9, is it also divisible by 3?

Yes! Since 9 = 3 times 3, any multiple of 9 is automatically a multiple of 3. But not vice versa: 12 is divisible by 3 but not by 9.

Glossary

Divisible: A number is divisible by another if dividing leaves no remainder. Example: 12 is divisible by 3.
Remainder: The amount left over after division. Example: 14 divided by 3 equals 4 remainder 2.
Factor: A number that divides evenly into another. Example: 3 is a factor of 12.
Even number: A number divisible by 2 (ends in 0, 2, 4, 6, or 8).
Digit sum: The result of adding all digits in a number. Example: digit sum of 456 is 4 + 5 + 6 = 15.

Formula Card

Divisible by 2

Last digit is 0, 2, 4, 6, or 8

Check if the ones digit is even

Divisible by 3

Sum of digits divisible by 3

Add all digits and test the sum

Divisible by 4

Last 2 digits divisible by 4

Test only the last two digits

Divisible by 5

Last digit is 0 or 5

Check the ones digit only

Divisible by 6

Divisible by 2 AND 3

Must pass both the 2 and 3 tests

Divisible by 9

Sum of digits divisible by 9