Divisibility Rules

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

Elementary25 minLesson
Printable worksheet

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.

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Which number is divisible by ?

Worked Examples

Is divisible by ?

1

Look at the last digit

The last digit of is → Last digit:

2

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

is an even number → Yes, is even

3

Apply the rule

If the last digit is even, the whole number is divisible by → Divisible by

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, ends in but is NOT divisible by .

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, is divisible by but not by , so it's not divisible by .

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: (digits sum to ) is divisible by but not by .

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, ends in but R .

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

Interactive Visual

Factor Tree

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Practice Problems

15 problems
Problem 1 of 15
Easy

Which number is divisible by ?

Why It Matters

Divisibility rules save time and mental energy:
  • Simplifying fractions: Quickly find common factors to reduce to
  • 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: ends in (divisible by ) and (divisible by ), so yes, divisible by . 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 , and . 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

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.
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.
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.
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

Add all digits and test the sum

Divisible by 10

Last digit is 0

Check if the number ends in zero

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