Greatest Common Factor Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Greatest Common Factor.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The greatest common factor (GCF) of two or more numbers is the largest positive integer that divides each of them evenly, with no remainder. It is also called the greatest common divisor (GCD).

The biggest 'piece' size that fits evenly into two numbers—like the largest tile that covers both a 12-unit and 18-unit floor.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The GCF is the largest whole number that divides every given number evenly.

Common stuck point: The procedure for greatest common factor is the easy part; the trap is picking the LCM by mistake. Asking "Am I looking for the largest number that divides every given value with no remainder?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I looking for the largest number that divides every given value with no remainder?

Worked Examples

Example 1

easy

Find the GCF of

48

and

36

Answer

12

First step

Prime-factor each number:

48 = 2^4 \times 3

and

36 = 2^2 \times 3^2

Full solution

2
Identify the common prime factors and keep the smaller exponent for each: $2^2$ and $3^1$ .
3
Multiply those shared factors: $2^2 \times 3 = 4 \times 3 = 12$ , so the GCF is $12$ .

The GCF is the product of shared prime factors, each raised to the lowest power appearing in either factorization. The GCF is useful for simplifying fractions.

Example 2

medium

Find the GCF of

84

126

, and

210

Example 3

easy

Find the GCF of

40

and

100

by prime factorization.

Example 4

medium

Use the Euclidean algorithm to find

\gcd(108, 60)

Example 5

medium

Use

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

to find

\gcd(15, 20)

given

\text{lcm}(15, 20) = 60

Example 6

medium

Find the GCF of

84

and

90

using the Euclidean algorithm.

Example 7

hard

Use the Euclidean algorithm to find

\gcd(252, 198)

Example 8

hard

A rectangular floor of

24

ft by

36

ft is to be tiled with identical square tiles, no cutting. Find the largest tile side length.

Example 9

hard

Find

\gcd(45, 75, 90)

Example 10

challenge

Find integers

x, y

such that

\gcd(56, 15) = 56x + 15y

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the GCF of

60

and

45

Example 2

easy

A teacher has ribbons of lengths

24

cm and

36

cm. She wants to cut them into the longest equal pieces with no leftover. How long should each piece be?

Example 3

easy

Find the GCF of

8

and

12

Example 4

easy

Find the GCF of

9

and

6

Example 5

easy

Find the GCF of

5

and

7

Example 6

easy

Find the GCF of

10

and

20

Example 7

easy

Find the GCF of

14

and

21

Example 8

easy

Find the GCF of

16

and

24

Example 9

easy

What is the GCF of any number

n

and itself?

Example 10

easy

Find the GCF of

18

and

24

Example 11

medium

Use prime factorization to find the GCF of

12

and

18

Example 12

medium

Find the GCF of

48

and

36

using prime factorization.

Example 13

medium

Find the GCF of

24

36

, and

60

Example 14

medium

Reduce

\frac{42}{56}

using the GCF.

Example 15

medium

Two ribbons are 18 cm and 30 cm. What is the longest equal piece length that divides both with no waste?

Example 16

medium

\gcd(a,b)=1

, what does that tell you about

a

and

b

Example 17

medium

Find the GCF of

7^2\cdot11

and

7\cdot11^2

Example 18

challenge

Given

\gcd(a,b)=6

and

\mathrm{lcm}(a,b)=72

, find

a\times b

Example 19

challenge

Use the Euclidean algorithm to find

\gcd(48,18)

Example 20

challenge

Find the largest positive integer

n

that divides both

n+12

and

n+20

Example 21

medium

Find the GCF of

30

and

45

using prime factorization.

Example 22

medium

Reduce

\frac{36}{48}

using the GCF.

Example 23

easy

Find the GCF of

12

and

18

Example 24

easy

Find the GCF of

25

and

35

Example 25

easy

Find the GCF of

30

and

50

Example 26

easy

Find the GCF of

32

and

48

Example 27

medium

Find the GCF of

72

and

96

Example 28

medium

Find the GCF of

144

and

216

Example 29

medium

A florist has

54

roses and

42

tulips. What is the greatest number of identical bouquets she can make using all the flowers?

Example 30

medium

Simplify

\dfrac{72}{96}

to lowest terms.

Example 31

medium

Find the GCF of

20

30

, and

50

Example 32

hard

\gcd(a, b) = 8

and

a \cdot b = 384

, find

\text{lcm}(a, b)

Example 33

hard

Find

\gcd(2^6 \times 3^4 \times 5, \, 2^4 \times 3^5 \times 7)

Example 34

hard

A box of

84

pens and a box of

112

pencils are split among children with each child getting the same number of pens and the same number of pencils, none left over. What is the largest possible number of children?

Example 35

challenge

For positive integers

a

and

b

with

\gcd(a,b)=g

and

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

, find

a+b

given

g=6

and

\ell=180

and

a \ge b

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Related Concepts

Factors Divisibility Intuition Simplification Least Common Multiple

Background Knowledge

These ideas may be useful before you work through the harder examples.

factorsdivisibility intuition