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Factors &
Multiples

Master the building blocks of numbers: learn to find factors, list multiples, and determine how numbers relate through divisibility.

Pre-Algebra

What Is a Factor?

A factor of a number is any whole number that divides into it evenly with no remainder. When you say "\( a \) is a factor of \( b \)," you mean \( b \div a = \text{whole number} \).

Factors of 12 3 rows 4 columns
\( 3 \times 4 = 12 \)

What Is a Multiple?

A multiple of a number is the result you get when you multiply that number by any whole number. If \( m = n \times k \) where \( k \) is a whole number, then \( m \) is a multiple of \( n \).

0 5 10 15 20 ...
\(+5\)
\(+5\)
\(+5\)
\(+5\)
 Multiples of 5: 5, 10, 15, 20, 25, ...

Why Do They Matter?

Simplifying Fractions
Finding common factors lets you reduce fractions like \(\frac{8}{12}\) to \(\frac{2}{3}\).
Adding Fractions
Common multiples give you a shared denominator: \(\frac{1}{3}+\frac{1}{4}\) needs LCM of 3 and 4.
Real Life
Arranging items in equal groups, scheduling events, and splitting things fairly all use divisibility.

The Divisibility Rule

A whole number \( a \) is a factor of \( b \) if and only if \( b \div a \) has remainder zero. Equivalently:

a is a Factor of b b is a Multiple of a same relationship

Finding All Factors: Factor Pairs

To find every factor of a number, list all factor pairs: two numbers that multiply to give the target. Stop when the pairs start repeating.

Factor Pairs of 24 1 2 3 4 6 8 12 24
\(1 \times 24\)
\(2 \times 12\)
\(3 \times 8\)
\(4 \times 6\)

Factors vs Multiples: Key Distinction

Factors Multiples
Direction Divide into the number Multiply out from the number
Count Finite (limited set) Infinite (never ends)
Size \(≤\) the number \(≥\) the number
Example for 6 1, 2, 3, 6 6, 12, 18, 24, ...
Operation \( b \div a = k \) \( a \times k = m \)

How to Find Factors: Step by Step

1
Start at 1. Every number has 1 and itself as factors.
2
Test each number: Try dividing by 2, then 3, then 4, and so on. If the division has no remainder, both the divisor and the quotient are factors.
3
Stop when pairs repeat. Once your test number meets or passes the quotient, you have found all factor pairs.
4
List in order. Write all unique factors from smallest to largest.

Example: Factors of 36

Systematic Factor Search Test Division Factor Pair 1
\( 36 \div 1 = 36 \)
 (1, 36) 2
\( 36 \div 2 = 18 \)
 (2, 18) 3
\( 36 \div 3 = 12 \)
 (3, 12) 4
\( 36 \div 4 = 9 \)
 (4, 9) 6
\( 36 \div 6 = 6 \)
 (6, 6) STOP

Example: First 8 Multiples of 7

Multiply 7 by the first 8 positive whole numbers:

\( 7 \times 1 = \color{#818cf8}7 \)
\( 7 \times 2 = \color{#818cf8}14 \)
\( 7 \times 3 = \color{#818cf8}21 \)
\( 7 \times 4 = \color{#818cf8}28 \)
\( 7 \times 5 = \color{#818cf8}35 \)
\( 7 \times 6 = \color{#818cf8}42 \)
\( 7 \times 7 = \color{#818cf8}49 \)
\( 7 \times 8 = \color{#818cf8}56 \)

Example: Is 8 a Factor of 56?

1
Divide: \( 56 \div 8 = 7 \)
2
Check the remainder: The quotient is exactly 7 with no remainder.
3
Conclusion: Since \( 8 \times 7 = 56 \), yes: 8 is a factor of 56, and 56 is a multiple of 8.

GCF and LCM: A Preview

Greatest Common Factor
The GCF of two numbers is the largest factor they share. Factors of 12: {1,2,3,4,6,12}. Factors of 18: {1,2,3,6,9,18}. GCF = 6.
Least Common Multiple
The LCM of two numbers is the smallest multiple they share. Multiples of 4: 4,8,12... Multiples of 6: 6,12... LCM = 12.
Factors of 12 Factors of 18 4 12 1 2, 3 6 9 18 GCF = 6

Divisibility Quick Tests

Divisor Quick Test Example
2 Last digit is even 48: last digit 8 ✓
3 Digit sum divisible by 3 123: \( 1+2+3 = 6 \) ✓
5 Last digit is 0 or 5 85: last digit 5 ✓
9 Digit sum divisible by 9 729: \( 7+2+9 = 18 \) ✓
10 Last digit is 0 150: last digit 0 ✓

Prime Factorization

A prime number has exactly two factors: 1 and itself. Every number can be broken down into a product of primes. This is called prime factorization.

Factor Tree: 60 60 6 10 2 3 2 5
\( 60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5 \)

Common Mistakes

✕ Confusing Factors and Multiples
Saying "12 is a factor of 3" when you mean "12 is a multiple of 3." Remember: factors divide into, multiples come from.
✕ Forgetting 1 and the Number Itself
Every number has 1 and itself as factors. Students often skip these when listing factors.
✕ Stopping the Factor Search Too Early
Testing only small numbers and missing factors like 9 in 36. Always test up to \(\sqrt{n}\) to catch all pairs.

Summary at a Glance

Concept Definition Example
Factor Divides evenly into a number 3 is a factor of 15
Multiple Result of multiplying by a whole number 15 is a multiple of 3
Factor Pair Two numbers whose product is the target (3, 5) for 15
Prime Exactly two factors: 1 and itself 2, 3, 5, 7, 11, 13
GCF Largest shared factor of two numbers \( \text{GCF}(12, 18) = 6 \)
LCM Smallest shared multiple of two numbers \( \text{LCM}(4, 6) = 12 \)

Question 1 of 5

Which of the following is NOT a factor of 42?

Question 2 of 5

Which number is a multiple of 9?

Question 3 of 5

How many factors does 28 have?

Question 4 of 5

If \( 15 \) is a factor of \( n \), which of the following must also be a factor of \( n \)?

Question 5 of 5

What is the least common multiple (LCM) of 6 and 8?