Teacher Guide: Combined Divisibility Rules

Combine simple rules to test for 12, 15, 18 and 36, learn the alternating sum rule for 11, and find every divisor from 2 to 12 in seconds.

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For Teachers

Learning Objectives

Combine divisibility rules for coprime factors to test for 12, 15, 18 and 36
Explain with a counterexample why rules for factors that are not coprime, such as 2 and 6, cannot be combined to test their product
Apply the alternating sum rule to test divisibility by 11
Find all divisors from 2 to 12 of a number quickly and solve missing digit problems under two rules

Prerequisites

• Divisibility rules for 2, 5 and 10 (lesson Divisibility by 2, 5, and 10)
• Digit sum rules for 3 and 9 (lesson Divisibility by 3 and 9)
• Rules for 4, 8 and 6 = 2 × 3 (lesson Divisibility by 4, 6, and 8)
• Factors, multiples and remainders

Discussion Starters

1. 18 is divisible by 2 and by 6. Why is it not divisible by 12?
2. Which pairs of numbers could you use to test for 24? Which pairs would give wrong answers?
3. Why does 11 need an alternating sum while 9 only needs the digit sum?
4. A number passes the tests for 4 and 9. Which other numbers must divide it?
5. Where in daily life would you want to know if a number divides by 12 or 15?

Common Misconceptions

Any two factors whose product is the divisor give a valid test (12 = 2 × 6)

Remediation: Let students test 18, 30 and 42 against 2 and 6, then divide by 12. Link the failure to the shared factor 2 and to lcm(2, 6) = 6.

An alternating sum of 0 means the number is not divisible by 11

Remediation: Ask what 0 ÷ 11 is. Show 1386: 6 - 8 + 3 - 1 = 0 and 1386 = 11 × 126.

If a number is divisible by 12, it must also be divisible by 24

Remediation: Contrast 36 (divisible by 12, not 24) with 48. Divisibility goes from the product to its factors, not the other way.

Differentiation Ideas

For Struggling Students:

• Give a two-column table: test 1, test 2, both passed?
• Start with 15 = 3 × 5 and 18 = 2 × 9, where both tests are easy
• Use three- and four-digit numbers only

For On-Level Students:

• Run full factor searches from 2 to 12 on four-digit numbers
• Solve missing digit problems for 12, 18 and 36
• Find and explain counterexamples for pairs that are not coprime

For Advanced Students:

• Build tests for 24, 45, 72 and 99 from coprime pairs
• Prove the rule for 11 with remainders of powers of 10
• Test 12-digit numbers for 11 and 12 and check with modular arithmetic

Standards Alignment

4.OA.B.4 (CCSS.MATH.CONTENT.4.OA.B.4)
Find all factor pairs for a whole number and determine whether it is a multiple of a given one-digit number
6.NS.B.4 (CCSS.MATH.CONTENT.6.NS.B.4)
Find the greatest common factor and the least common multiple of two whole numbers

Lesson Resources

visualDivisibility Practice Board
Students guess yes or no for every divisor from 2 to 12, then reveal each rule and its verdict
activityFind the Counterexample
Pairs hunt for numbers that pass two tests but fail the product (2 and 6, 4 and 6, 3 and 12) and explain why
gameMissing Digit Race
Teams race to find the digit that makes a number divisible by 12, 18 or 36
worksheetFactor Search Grid
A grid of numbers to test against 2 to 12, including 11 with the alternating sum

Lesson Content

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

Definition

Some divisors have no rule of their own, but you can combine two simple rules.

Combined rule: if

a

and

b

are coprime (their only common factor is

1

), then a number divisible by both

a

and

b

is divisible by

a \times b

Divisor	Test	Why it works
$12 = 3 \times 4$	digit sum divisible by 3 and last two digits divisible by 4	3 and 4 are coprime
$15 = 3 \times 5$	digit sum divisible by 3 and last digit 0 or 5	3 and 5 are coprime
$18 = 2 \times 9$	last digit even and digit sum divisible by 9	2 and 9 are coprime
$36 = 4 \times 9$	last two digits divisible by 4 and digit sum divisible by 9	4 and 9 are coprime

Careful:

12 = 2 \times 6

is NOT a valid test, because 2 and 6 share the factor 2. The number

18

is divisible by 2 and by 6, but

18 \div 12 = 1.5

. In general, passing the tests for

a

and

b

proves divisibility by

\text{lcm}(a, b)

, which equals

a \times b

only when

a

and

b

are coprime.

Divisibility by 11 (alternating sum): start at the ones digit and add and subtract the digits in turn:

+ - + - \dots

If the result is divisible by 11 (this includes

0

11

-11

22

), the number is divisible by 11.

82{,}918: \quad 8 - 1 + 9 - 2 + 8 = 22 \Rightarrow 82{,}918 \text{ is divisible by } 11

Worked Examples

7128

divisible by

12

Split 12 into coprime factors

12 = 3 \times 4

, and 3 and 4 have no common factor except 1 → Test 3 and 4

Test 3 with the digit sum

7 + 1 + 2 + 8 = 18

, and

18

is divisible by 3

Test 4 with the last two digits

The last two digits form

28

, and

28 \div 4 = 7

→ Divisible by 4

Combine and check

Both tests pass, so 12 divides it:

7128 \div 12 = 594

→ Divisible by 12

Answer: Yes, $7128$ is divisible by $12$ ( $7128 = 12 \times 594$ ).

Common Mistakes

Testing 12 with the rules for 2 and 6

Why it's wrong: 2 and 6 share the factor 2, so passing both only proves divisibility by 6. Counterexample: $18$ (and also $30$ ) is divisible by 2 and 6 but not by 12.

Correct: Use coprime factors: $12 = 3 \times 4$ . Check the digit sum for 3 and the last two digits for 4.

Testing 36 with 3 and 12

Why it's wrong: 3 and 12 share the factor 3. $24$ is divisible by 3 and by 12, but $24 \div 36$ is not a whole number.

Correct: Use $36 = 4 \times 9$ : last two digits divisible by 4 and digit sum divisible by 9.

Thinking one passed test is enough

Why it's wrong: A combined rule needs BOTH tests. $25$ ends in 5 but its digit sum is 7, so it is not divisible by 15.

Correct: Check every factor of the pair. If one test fails, the number is not divisible by the product.

Reading an alternating sum of $0$ or $-11$ as not divisible by 11

Why it's wrong: $0$ , $-11$ and $-22$ are multiples of 11 too. For $9174$ : $4 - 7 + 1 - 9 = -11$ , and indeed $9174 = 11 \times 834$ .

Correct: The number is divisible by 11 whenever the alternating sum is $0$ or any multiple of 11, positive or negative.

Why It Matters

Combined rules turn big questions into two quick checks:

Sharing equally: can 1350 stickers go to 15 classes with none left over? Two checks say yes.
Timetables: a day has 1440 minutes, so a tram every 36 minutes fits the day exactly if 1440 is divisible by 36.
Fractions: spotting that 12 or 18 divides the top and the bottom simplifies a fraction in one step.
Huge numbers: a 12-digit number can be tested for 11 or 12 in seconds, even where a calculator shows too few digits.

The idea behind every combined rule, coprime factors, is also the key to the least common multiple and to many puzzles with missing digits.

Real World Applications

Sharing Equally

Teachers, coaches and organizers often need to know whether items can be shared out with nothing left over.

Example:

A school has 1350 stickers for 15 classes. $15 = 3 \times 5$ : the last digit is 0 (divisible by 5) and $1 + 3 + 5 + 0 = 9$ (divisible by 3). So the stickers can be shared equally.

1Try It Yourself

A school wants to give 1350 stickers to 15 classes, the same number to each class.

How many stickers does each class get?

Step 1: Write the mathematical expression

Write the division that gives the stickers per class:

Timetables

Transport planners want departures that repeat in the same pattern every day. A day has $24 \times 60 = 1440$ minutes.

Example:

A bus every 12 minutes fits a day exactly: $1 + 4 + 4 + 0 = 9$ (divisible by 3) and $40 \div 4 = 10$ , so 1440 is divisible by 12, giving 120 departures.

2Try It Yourself

A tram leaves every 36 minutes, all day and all night, starting at midnight.

How many trams leave in one day of 1440 minutes?

Step 1: Write the mathematical expression

Write the division that gives the number of departures:

The 12-Digit Check

Many calculators show only 8 to 10 digits, so they cannot divide a 12-digit number exactly. Divisibility rules work on numbers of any length.

Example:

For $123{,}456{,}789{,}012$ the digit sum is $48$ (divisible by 3) and the last two digits form $12$ (divisible by 4), so the number is divisible by 12.

3Try It Yourself

A 12-digit code is $123{,}456{,}789{,}012$ . You want to know if it is divisible by 12 and by 11, without a calculator.

What is the remainder when $123{,}456{,}789{,}012$ is divided by 12?

Step 1: Write the mathematical expression

Enter the remainder:

Key Takeaways

1If $a$ and $b$ are coprime, a number divisible by both $a$ and $b$ is divisible by $a \times b$
2Test 12 with 3 and 4, 15 with 3 and 5, 18 with 2 and 9, 36 with 4 and 9
3Never test a product with two factors that share a factor: 18 is divisible by 2 and 6 but not by 12
4Divisible by 11: the alternating sum of the digits ( $+ - + - \dots$ from the ones digit) is a multiple of 11, including $0$ and negative multiples
5For a quick factor search, use last-digit tests, digit sums and the alternating sum, then combine coprime pairs
6For a missing digit under two rules, list the digits that pass each rule and keep the ones in both lists

Frequently Asked Questions

Why must the two factors be coprime?

A number divisible by

a

and by

b

is a common multiple of

a

and

b

, so it is divisible by

\text{lcm}(a, b)

. When

a

and

b

are coprime,

\text{lcm}(a, b) = a \times b

. When they share a factor, the lcm is smaller:

\text{lcm}(2, 6) = 6

, not 12.

Why does the alternating sum work for 11?

10 = 11 - 1

, so 10 leaves the remainder

-1

when divided by 11,

100 = 99 + 1

leaves

+1

1000 = 1001 - 1

leaves

-1

, and so on. So each digit counts with the sign

+, -, +, -, \dots

from the ones digit, and the number has the same remainder as its alternating sum.

Does it matter whether I start the alternating sum from the left or the right?

For the yes or no answer it does not: the two results are equal or opposite numbers, and

-22

is a multiple of 11 exactly when

22

is. Starting at the ones digit with

+

also keeps the remainder of the number (add or subtract 11 until you land between 0 and 10), so it is the safer habit.

Can I test for 24 with 4 and 6?

No, 4 and 6 share the factor 2.

12

is divisible by 4 and 6 but not by 24. Use the coprime pair

24 = 3 \times 8

: digit sum divisible by 3 and last three digits divisible by 8.

Glossary

Coprime: Two numbers are coprime if their only common factor is 1. Example: 4 and 9 are coprime, 2 and 6 are not.
Combined divisibility rule: A test for a product $a \times b$ of coprime numbers: the number must pass the test for $a$ and the test for $b$ .
Alternating sum: The digits added and subtracted in turn from the ones digit: for 2816 it is $6 - 1 + 8 - 2 = 11$ .
Digit sum: The sum of all digits of a number. The digit sum of 1980 is $1 + 9 + 8 + 0 = 18$ .
Remainder: What is left over after dividing. $18 \div 12 = 1$ remainder $6$ .
Least common multiple (lcm): The smallest number that is a multiple of both numbers. $\text{lcm}(3, 4) = 12$ , $\text{lcm}(2, 6) = 6$ .

Formula Card

Combined rule

a, b

coprime:

a \mid n

and

b \mid n \Rightarrow a \times b \mid n

Works only when the two factors share no factor except 1

Divisible by 12

12 = 3 \times 4

Digit sum divisible by 3 and last two digits divisible by 4

Divisible by 15

15 = 3 \times 5

Digit sum divisible by 3 and last digit 0 or 5

Divisible by 18

18 = 2 \times 9

Last digit even and digit sum divisible by 9

Divisible by 36

36 = 4 \times 9

Last two digits divisible by 4 and digit sum divisible by 9

Divisible by 11

d_0 - d_1 + d_2 - d_3 + \dots

Alternating sum from the ones digit is a multiple of 11 (including 0)

Teacher Guide: Combined Divisibility Rules

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Sharing Equally

Timetables

The 12-Digit Check

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Combined rule

Divisible by 12

Divisible by 15

Divisible by 18

Divisible by 36

Divisible by 11

More in This Topic

Divisibility Rules

Divisibility by 2, 5, and 10

Divisibility by 3 and 9

Divisibility by 4, 6, and 8