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Math Lesson 1.5.7 - Greatest Common Factor, GCF

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Welcome to our Math lesson on Greatest Common Factor, GCF, this is the seventh lesson of our suite of math lessons covering the topic of Multiples, Factors, Prime Numbers and Prime Factorization including LCM and GCF, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Greatest Common Factor, GCF

Another very important thing to identify and calculate in a set of two or more numbers is the greatest common factor (GCF). It represents the greatest number by which both original numbers are divided. For example, GCF of 8 and 12 is 4 because both 8 and 12 are divisible by 4. Although these two numbers are divisible by 2 as well, we are interested in the greatest divisor (factor).

We can use the tabular method again - the same used to calculate the LCM, but this time we consider only the prime factors that divide all numbers simultaneously. Then, the greatest common factor, GCF, is calculated by multiplying these common prime factors. Look at the following example.

Example 8

Calculate the least common multiple (LCM) and greatest common factor (GCF) of 48 and 72.

Solution 8

We use the tabular method to find both LCM and GCF. The common factors are circled and then we multiply them to find the GCF. Thus,

Math Tutorials: Multiples, Factors, Prime Numbers and Prime Factorization including LCM and GCF Example

The LCM of 48 and 72 therefore is

LCM (48, 72) = 24 × 32 = 16 × 9 = 144

The GCF of 48 and 72 (only the circled factors) therefore is

LCM (48, 72) = 23 × 3 = 8 × 3 = 24

We can extend this method to more than two numbers, as we did for LCM. Look at the example below.

Example 9

Calculate the LCM and GCF of 64, 80 and 120.

Solution 9

Again, we use the tabular method for finding both LCM and GCF simultaneously. We have

Math Tutorials: Multiples, Factors, Prime Numbers and Prime Factorization including LCM and GCF Example

Hence, LCM (64, 80, 120) = 26 × 3 × 5 = 64 × 3 × 5 = 960, and GCF (64, 80, 120) = 23 = 8.

Remark! If one number is a multiple of another, their LCM is the biggest number while their GCF is the smallest one. For example, LCM (4, 12) = 12 and GCF (4, 12) = 4.

The concept of greatest common factor is also very useful in practice. Let's see an example to illustrate this point.

Example 10

A 56 m × 40 m rectangular field has to be surrounded by wire fence. To make the fence stable, we must fix vertical poles to the ground and tie the fence to them. What is the minimum number of poles required if they must be equidistant from each other?

Solution 10

Using the minimum number of poles means placing them at the largest possible distance from each other. Since poles have the same distance from each other, we must find the greatest common factor of the two dimensions, i.e. of 40 and 56, which represents the distance between poles. We have

Math Tutorials: Multiples, Factors, Prime Numbers and Prime Factorization including LCM and GCF Example

Thus, we have GCF (40, 56) = 23 = 8. This means the poles must be 8 m away from each other. Therefore, since the field's perimeter (i.e. its total length) is P = 2 × 40 m + 2 × 56 m = 80 m + 112 m = 192 m, we obtain for the minimum number of poles required:

Minimum number of poles = Total length of fence/Distance between poles
= 192 m/8 m
= 24 poles

More Multiples, Factors, Prime Numbers and Prime Factorization including LCM and GCF Lessons and Learning Resources

Arithmetic Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
Tutorial
Revision
Notes
Revision
Questions
1.5Multiples, Factors, Prime Numbers and Prime Factorization including LCM and GCF
Lesson IDMath Lesson TitleLessonVideo
Lesson
1.5.1What are Multiples of a Number? What are Factors?
1.5.2Finding the Factors of a Number Using the Tree Method
1.5.3Prime Numbers
1.5.4Prime Factorization
1.5.5Finding the Common Multiples of Two Numbers
1.5.6Finding the LCM of Two or More Numbers
1.5.7Greatest Common Factor, GCF

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