Teacher Guide: Divisibility by 3 and 9

Learn the digit sum rule: add the digits of a number to see at once whether it is divisible by 3 or by 9, find missing digits and check your sums.

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For Teachers

Learning Objectives

Find the digit sum and the repeated digit sum of a whole number
Use the digit sum to decide whether a number is divisible by 3 and by 9
Explain why every multiple of 9 is a multiple of 3, but not the other way round
Explain the rule with place value: 10 = 9 + 1, 100 = 99 + 1, 1000 = 999 + 1
Find a missing digit so that a number is divisible by 3 or 9, and check a sum by casting out nines

Prerequisites

• Divisibility Rules: what divisible by means and what a remainder is
• Divisibility by 2, 5, and 10: rules that look at the last digit
• Multiplication facts of 3 and 9
• Place value: ones, tens, hundreds, thousands

Discussion Starters

1. Why can the last digit tell us about 2, 5 and 10, but not about 3 and 9?
2. If you swap the digits of a number divisible by 9, is the new number still divisible by 9? Why?
3. Can you find a number divisible by 3 but not by 9? Can you find one divisible by 9 but not by 3?
4. What is special about 10 = 9 + 1? Would a similar trick work for 11?
5. Casting out nines says an answer is fine. Can it still be wrong?

Common Misconceptions

A number ending in 3, 6 or 9 is divisible by 3

Remediation: Show counterexamples side by side: 13, 16, 19 are not divisible by 3, but 12, 15, 21 are. Then add the digits of each to show the digit sum is what decides.

Divisible by 3 and divisible by 9 mean the same thing

Remediation: List multiples of 3 up to 36 and circle the multiples of 9: every third one is circled. Digit sums 3, 6 give 3 only, digit sum 9 gives both.

The digit sum is the answer to the division

Remediation: Stress that the digit sum only answers yes or no (and gives the remainder). For 4527 the digit sum is 18, but 4527 ÷ 9 is 503.

A matching casting-out-nines check means the answer is surely right

Remediation: Give 2345 + 1678 = 4032 (digits swapped). The check passes, yet the answer is wrong. The check catches many slips, not all.

Differentiation Ideas

For Struggling Students:

• Start with 2-digit numbers and the times tables of 3 and 9 on the desk
• Use digit cards or counters to add the digits one by one
• Check every answer with a calculator so students trust the rule

For On-Level Students:

• Use 4- to 6-digit numbers and repeated digit sums
• Solve missing digit puzzles for 3 and for 9
• Check additions by casting out nines

For Advanced Students:

• Prove the rule for a 3-digit number abc using 100a + 10b + c
• Find all 3-digit numbers that start with 1 and are divisible by 9, and explain the pattern
• Combine rules: which digits make 7_2 divisible by 6 (by 2 and by 3)?

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 and determine whether a given whole number is a multiple of a given one-digit number
4.NBT.A.1 (CCSS.MATH.CONTENT.4.NBT.A.1)
Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right
6.NS.B.4 (CCSS.MATH.CONTENT.6.NS.B.4)
Find the greatest common factor and least common multiple of whole numbers and use the distributive property

Lesson Resources

visualDivisibility Practice
Students guess yes or no for 3 and 9, then the tool reveals the digit sum and the real division
visualWhy It Works
A long number split into multiples of 999, 99 and 9 plus its digit sum
activityDigit Card Shuffle
Pairs build 3- and 4-digit numbers from digit cards, predict divisibility by 3 and 9, then shuffle the cards and see that the answer stays the same
gameMissing Digit Race
Teams race to find every digit that makes a number like 52_4 divisible by 3, then the one digit for 9
worksheetCheck the Sums
Ten additions, some with mistakes: students cast out nines to find the wrong ones

Lesson Content

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

Definition

To test whether a number is divisible by 3 or divisible by 9, the last digit does not help. Instead, you add up all of its digits. This total is called the digit sum.

The rule for 3

A number is divisible by

3

when its digit sum is divisible by

3

516: \quad 5 + 1 + 6 = 12, \quad 12 \div 3 = 4 \quad \checkmark \quad (516 \div 3 = 172)

The rule for 9

A number is divisible by

9

when its digit sum is divisible by

9

4527: \quad 4 + 5 + 2 + 7 = 18, \quad 18 \div 9 = 2 \quad \checkmark \quad (4527 \div 9 = 503)

Big numbers: add again

If the digit sum is still big, add its digits again until one digit is left:

987{,}654 \rightarrow 39 \rightarrow 12 \rightarrow 3

The last digit you reach tells you everything:

3

6

9

means divisible by

3

, and only

9

means divisible by

9

9 is stricter than 3

Every number divisible by

9

is also divisible by

3

(because

9 = 3 \times 3

). The reverse is not true:

516

is divisible by

3

, but its digit sum

12

is not divisible by

9

, so

516

is not divisible by

9

Worked Examples

516

divisible by

3

? Is it divisible by

9

Add all the digits

5 + 1 + 6 = 12

→ Digit sum:

12

Test the digit sum for 3

12 \div 3 = 4

, no remainder →

516

is divisible by

3

516 \div 3 = 172

Test the digit sum for 9

12 \div 9 = 1

remainder

3

→

516

is not divisible by

9

Answer: $516$ is divisible by $3$ ( $516 \div 3 = 172$ ) but not by $9$ , because its digit sum $12$ is a multiple of $3$ but not of $9$ .

Common Mistakes

Looking at the last digit, as for 2, 5 and 10

Why it's wrong: The last digit says nothing about 3 or 9. $23$ ends in $3$ but is not divisible by $3$ , while $27$ ends in $7$ and is divisible by $3$ and by $9$ .

Correct: Always add all the digits: $2 + 7 = 9$ , so $27$ is divisible by $3$ and by $9$ .

Thinking every number divisible by 3 is also divisible by 9

Why it's wrong: The rule for $9$ is stricter. A digit sum of $12$ or $15$ is a multiple of $3$ but not of $9$ .

Correct: $516$ (digit sum $12$ ) is divisible by $3$ but not by $9$ . The other way round always works: every multiple of $9$ is a multiple of $3$ .

Stopping after the first digit that works in a missing digit puzzle

Why it's wrong: For $3$ there are usually three or four digits that work, because the multiples of $3$ are only $3$ apart.

Correct: For $52\square4$ and $3$ , check $12$ , $15$ and $18$ : the box can be $1$ , $4$ or $7$ .

Believing that a matching digit sum check proves the answer is right

Why it's wrong: Casting out nines only finds some mistakes. Swapping two digits does not change the digit sum, so $4032$ passes the same check as $4023$ .

Correct: If the check does not match, the answer is surely wrong. If it matches, the answer is probably right, but not certainly.

Why It Matters

The digit sum rule is one of the most useful shortcuts in arithmetic:

Sharing fairly: Can 378 stickers be split among 9 friends? $3 + 7 + 8 = 18$ , so yes, without dividing first.
Simplifying fractions: In $\frac{51}{87}$ both digit sums ( $6$ and $15$ ) are divisible by $3$ , so you can divide both by $3$ : $\frac{17}{29}$ .
Checking your work: The digit sum keeps the remainder after dividing by $9$ , so you can check a long addition in a few seconds (this is called casting out nines).
Number puzzles: Missing digit puzzles and codes often hide a number divisible by $3$ or $9$ .

After the rules for 2, 5 and 10 (which look at the last digit), this is the first rule that looks at every digit, and it prepares you for the rules for 6 and 12.

Real World Applications

Making Equal Teams

Coaches and teachers often need to split a group into equal teams. The digit sum tells them at once whether it works.

Example:

Can $135$ runners start in $9$ equal groups? $1 + 3 + 5 = 9$ , so yes: $135 \div 9 = 15$ runners per group.

1Try It Yourself

A school sports day has $378$ students. The teachers want $9$ teams of the same size.

How many students will be in each team?

Step 1: Write the mathematical expression

Divide the students into 9 teams:

Checking a Bill

Before calculators, shopkeepers and bookkeepers checked long additions with digit sums. It is still a quick way to catch a slip in your own calculations.

Example:

$1236 + 2458 = 3694$ : the remainders are $3$ and $1$ , which add to $4$ , and $3694$ also gives $3 + 6 + 9 + 4 = 22 \rightarrow 4$ . The check passes.

2Try It Yourself

A cashier adds two delivery notes: $1236 + 2458 = 3694$ . You want to check the total with remainders after dividing by $9$ .

What remainder does the total $3694$ leave when divided by $9$ ?

Step 1: Write the mathematical expression

Find the remainder of 3694 divided by 9:

Cracking a Code

Puzzle books and escape rooms like codes with a missing digit. A divisibility clue often gives the answer in one step.

Example:

A code $7\square2$ is divisible by $9$ : $7 + 2 = 9$ , so the box must be $0$ or $9$ ( $702$ or $792$ ).

3Try It Yourself

The code of a bike lock is $4\square71$ . A note says: the code is divisible by $9$ .

Which digit is missing?

Step 1: Write the mathematical expression

Find the digit that brings the digit sum to the next multiple of 9:

Key Takeaways

1The digit sum is the total of all digits of a number: for $4527$ it is $4 + 5 + 2 + 7 = 18$
2A number is divisible by $3$ when its digit sum is divisible by $3$ , and by $9$ when its digit sum is divisible by $9$
3For big numbers, add the digits again and again until one digit is left: $3$ , $6$ , $9$ mean divisible by $3$ , only $9$ means divisible by $9$
4Every multiple of $9$ is a multiple of $3$ , but not every multiple of $3$ is a multiple of $9$
5It works because $10 = 9 + 1$ , $100 = 99 + 1$ , $1000 = 999 + 1$ : the nines drop out and only the digit sum is left
6A number and its digit sum leave the same remainder when divided by $3$ or $9$ , which lets you find missing digits and check sums (casting out nines)

Frequently Asked Questions

Does the order of the digits matter?

No.

4527

7254

and

2745

all have digit sum

18

, so all three are divisible by

9

. Moving digits around never changes the digit sum.

What about the digit 0?

0

adds nothing to the digit sum, so you can skip it.

3051

and

351

have the same digit sum

9

, and both are divisible by

9

Why does this trick not work for 7?

It works for

3

and

9

because

10

is exactly

1

more than a multiple of

9

. For

7

that is not true (

10 = 7 + 3

), so the place values do not drop out. Seven needs a different test.

How do I get the remainder from the digit sum?

Reduce the number to one digit. For

9

: that digit is the remainder, except that

9

means remainder

0

. For example

7654 \rightarrow 22 \rightarrow 4

, and indeed

7654 = 9 \times 850 + 4

Glossary

Digit sum: The total you get by adding all the digits of a number, e.g. $372 \rightarrow 3 + 7 + 2 = 12$
Repeated digit sum: Adding the digits again and again until only one digit is left, e.g. $987{,}654 \rightarrow 39 \rightarrow 12 \rightarrow 3$
Divisible by: A number is divisible by $3$ if dividing it by $3$ leaves no remainder, e.g. $372 \div 3 = 124$
Divisibility rule: A shortcut that tells whether a number is divisible by another without doing the division
Remainder: What is left over after dividing, e.g. $23 \div 9 = 2$ remainder $5$
Casting out nines: Checking a calculation by replacing every number with its remainder after dividing by $9$ (its repeated digit sum)

Formula Card

Rule for 3

3 \mid n \iff 3 \mid \text{digit sum of } n

A number is divisible by 3 when its digit sum is divisible by 3

Rule for 9

9 \mid n \iff 9 \mid \text{digit sum of } n

A number is divisible by 9 when its digit sum is divisible by 9

Why it works

4527 = 4 \times 999 + 5 \times 99 + 2 \times 9 + (4 + 5 + 2 + 7)

The parts with 999, 99 and 9 are always multiples of 9, so only the digit sum decides

Same remainder

7654 \rightarrow 22 \rightarrow 4, \quad 7654 = 9 \times 850 + 4

A number leaves the same remainder as its digit sum when divided by 9 (or 3)

Teacher Guide: Divisibility by 3 and 9

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

The rule for 3

The rule for 9

Big numbers: add again

9 is stricter than 3

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Making Equal Teams

Checking a Bill

Cracking a Code

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Rule for 3

Rule for 9

Why it works

Same remainder

More in This Topic

Divisibility Rules

Divisibility by 2, 5, and 10

Divisibility by 4, 6, and 8

Combined Divisibility Rules