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Divisibility
and Factors

Use divisibility rules and factors to reason about whole numbers quickly and confidently.

ISEE Lower Level

What Does Divisible Mean?

A number is divisible by another number when it divides evenly, leaving no remainder. For example, \( 12 \div 3 = 4 \) with nothing left over, so 12 is divisible by 3.

12 dots split into 3 equal groups:

What Is a Factor?

A factor of a number is a whole number that divides it evenly. Factors always come in pairs that multiply together to make the number.

Factor pairs of 12
\( 1 \times 12 \)
\( 2 \times 6 \)
\( 3 \times 4 \)
 Factors of 12: 1, 2, 3, 4, 6, 12

Why Factors Matter

Sharing Evenly
Factors tell you how to split items into equal groups with none left over.
Simplifying
Knowing factors helps reduce fractions to lowest terms.
Test Speed
Divisibility rules let you skip long division and answer fast.

Divisibility Rules: 2, 5, 10

2
Divisible by 2: the last digit is even (0, 2, 4, 6, or 8). Example: 134.
5
Divisible by 5: the last digit is 0 or 5. Example: 245.
10
Divisible by 10: the last digit is 0. Example: 470.

Divisibility Rules: 3 and 9

3
Divisible by 3: add up all the digits. If the sum is divisible by 3, so is the number.
9
Divisible by 9: add up all the digits. If the sum is divisible by 9, so is the number.

Even vs Odd

An even number is divisible by 2. An odd number is not. The last digit always tells you which.

EvenOdd
Last digit0, 2, 4, 6, 81, 3, 5, 7, 9
Divisible by 2?YesNo
Examples48, 130, 25637, 91, 205

Factors vs Multiples

FactorMultiple
DefinitionDivides evenly INTO a numberResult of multiplying a number
SizeLess than or equal to the numberGreater than or equal to the number
For 61, 2, 3, 66, 12, 18, 24

How to Find All Factors

1
Start at 1. Every number has 1 and itself as factors.
2
Test in order: 2, 3, 4... Check if each divides evenly.
3
Write factor pairs. When one divides evenly, record both numbers.
4
Stop in the middle. Once the pairs meet, you've found them all.

Example 1: Factors of 18

Factor pairs of 18
\( 1 \times 18 \)
\( 2 \times 9 \)
\( 3 \times 6 \)
 Factors: 1, 2, 3, 6, 9, 18

Example 2: Is 435 Divisible by 3?

Use the divisibility rule for 3: add up the digits and check the sum.

\( 4 + 3 + 5 = 12 \)
 Is 12 divisible by 3?
\( 12 \div 3 = 4 \)   ✓ Yes!

Example 3: Common Factors

A teacher has 24 pencils and 36 erasers. What group sizes split both evenly? Find the common factors.

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 Common factors: 1, 2, 3, 4, 6, 12

Common Mistakes

✕ Mixing Up Terms
Confusing factors (divide into) with multiples (built from). Remember: factors are small, multiples are big.
✕ Forgetting 1 and Itself
Every number's factor list must include both 1 and the number itself.
✕ Wrong Digit Sum
Skipping a digit when adding for the 3 or 9 rule gives a wrong answer.

Divisibility Rule Summary

Divisible byQuick TestExample
\(2\)Last digit is even58 ✓
\(3\)Digit sum divisible by 372 → \(7+2=9\) ✓
\(5\)Last digit 0 or 595 ✓
\(9\)Digit sum divisible by 9189 → \(1+8+9=18\) ✓
\(10\)Last digit is 0340 ✓

Question 1 of 5

Which number is divisible by 5?

Question 2 of 5

Which of these is a factor of 24?

Question 3 of 5

Use the digit-sum rule: which number is divisible by 3?

Question 4 of 5

How many factors does 16 have in total?

Question 5 of 5

Which number is a common factor of both 12 and 18?

You've Got the Tools!

You can now use divisibility rules to test numbers fast, find all the factors of a number, and spot common factors between numbers.