Arithmetic

Factors, Multiples, and Divisibility

Last updated: March 2026 · Beginner

Before you start

You should be comfortable with:

Factors and multiples are two sides of the same coin. If $3 \times 4 = 12$ , then 3 and 4 are both factors of 12, and 12 is a multiple of both 3 and 4. These concepts are essential for simplifying fractions, finding common denominators, and working with ratios.

Factors

A factor of a number divides into it evenly (with no remainder).

Example 1: Find all factors of 24

Systematically check which numbers divide 24 evenly, starting from 1:

Division	Factor pair
$24 \div 1 = 24$	1 and 24
$24 \div 2 = 12$	2 and 12
$24 \div 3 = 8$	3 and 8
$24 \div 4 = 6$	4 and 6
$24 \div 5 = 4.8$	Not a factor

Stop when your factors start repeating (when you pass the square root).

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Example 2: Find all factors of 36

Factor pair
1 and 36
2 and 18
3 and 12
4 and 9
6 and 6

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Multiples

A multiple of a number is what you get when you multiply it by 1, 2, 3, 4, and so on. Unlike factors (which are finite), multiples go on forever.

Example 3: List the first 8 multiples of 7

$7, 14, 21, 28, 35, 42, 49, 56$

Example 4: Is 45 a multiple of 9?

$45 \div 9 = 5$ (divides evenly) → Yes, 45 is a multiple of 9.

Example 5: Is 50 a multiple of 7?

$50 \div 7 = 7.14...$ (does not divide evenly) → No, 50 is not a multiple of 7.

Divisibility Rules

These shortcuts let you quickly check whether a number is divisible by a given factor without doing long division:

Divisible by	Rule	Example
2	Last digit is even (0, 2, 4, 6, 8)	4,538 → last digit 8 ✓
3	Sum of digits is divisible by 3	726 → $7+2+6=15$ , $15 \div 3 = 5$ ✓
4	Last two digits form a number divisible by 4	1,316 → 16 ÷ 4 = 4 ✓
5	Last digit is 0 or 5	2,345 → last digit 5 ✓
6	Divisible by both 2 AND 3	312 → even AND $3+1+2=6$ ✓
8	Last three digits form a number divisible by 8	5,120 → 120 ÷ 8 = 15 ✓
9	Sum of digits is divisible by 9	5,481 → $5+4+8+1=18$ , $18 \div 9 = 2$ ✓
10	Last digit is 0	7,830 → last digit 0 ✓

Example 6: Is 2,340 divisible by 2, 3, 4, 5, 6, 9, and 10?

By 2? Last digit is 0 (even) → Yes
By 3? $2+3+4+0 = 9$ , and $9 \div 3 = 3$ → Yes
By 4? Last two digits: 40. $40 \div 4 = 10$ → Yes
By 5? Last digit is 0 → Yes
By 6? Divisible by both 2 and 3 → Yes
By 9? Digit sum is 9, and $9 \div 9 = 1$ → Yes
By 10? Last digit is 0 → Yes

Factor vs. Multiple: How to Keep Them Straight

	Factors	Multiples
Definition	Numbers that divide into it	Numbers you get by multiplying
Count	Finite	Infinite
Size	Always ≤ the number	Always ≥ the number
Example (12)	1, 2, 3, 4, 6, 12	12, 24, 36, 48, 60, …

A helpful memory trick: Factors are Few, Multiples are Many.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Problem 1: Find all factors of 30

1, 2, 3, 5, 6, 10, 15, 30

Problem 2: List the first 6 multiples of 8

8, 16, 24, 32, 40, 48

Problem 3: Is 531 divisible by 3?

Sum of digits: $5 + 3 + 1 = 9$ . Since $9 \div 3 = 3$ , yes, 531 is divisible by 3.

Problem 4: Is 2,724 divisible by 4?

Last two digits: 24. $24 \div 4 = 6$ . Yes, 2,724 is divisible by 4.

Problem 5: Find all factors of 48

Factor pairs: (1, 48), (2, 24), (3, 16), (4, 12), (6, 8)

Factors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Key Takeaways

Factors divide into a number evenly; multiples are products of a number
Factors are finite and always ≤ the number; multiples are infinite and always ≥ the number
Divisibility rules let you quickly check common factors without division
Find factor pairs systematically — start from 1 and stop when pairs repeat
These concepts are the foundation for GCF and LCM, simplifying fractions, and finding common denominators

Return to Arithmetic for more foundational math topics.

Next Up in Arithmetic

Multiplying Whole Numbers Absolute Value Adding and Subtracting Whole Numbers Adding and Subtracting Decimals

All Arithmetic topics

Last updated: March 29, 2026