problem solving using gcf and lcm

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GCF and LCM word problems

Factors and multiples.

These word problems need the use of greatest common factors (GCFs) or least common multiples (LCMs) to solve. Mixing GCF and LCM word problems encourages students to read and think about the questions, rather than simply recognizing a pattern to the solutions.

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GCF and LCM

Here you will learn about GCF and LCM (greatest common factor and least common multiple), including how to find the GCF and LCM of two or more numbers using the prime factorization method and recognize when to find the GCF or the LCM in word problems.

Students will first learn about GCF and LCM as part of the number system in 6th grade.

What is GCF and LCM?

GCF and LCM are two abbreviations for the greatest common factor (GCF) and the least common multiple (LCM).

• The GCF is the largest whole number that two or more numbers can be divided by. Other names for this include the greatest common divisor (GCD) and the highest common factor (HCF). For example, find the GCF of 8 and 12. Let’s start by writing the factors of 8 and 12. Factors of {\bf{8}} {\textbf{: }} 1, 2, 4, 8 Factors of {\bf{12}} {\textbf{: }} 1, 2, 3, 4, 6, 12 There are several numbers that occur in both lists ( 1, 2, and 4 ). The largest factor that occurs in each list is 4, and so the greatest common factor of 8 and 12 is \bf{4}.
• The LCM is the smallest whole number that is a multiple of two or more whole numbers (exists within the multiplication table of each number). Another name for this is the lowest common multiple. For example, find the LCM of 8 and 12. Let’s start by writing the first 12 multiples of 8 and 12. Multiples of {\bf{8}} {\textbf{: }} 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96 Multiples of {\bf{12}} {\textbf{: }} 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144 There are several values that occur in both lists ( 24, 48, 72, and 96 ). The lowest of these is 24, hence the least common multiple of 8 and 12 is \bf{24}.

Prime factor decomposition

To calculate the GCF or LCM of two or more numbers, you can write out a list of factors or multiples as we have above, however, this approach can be very time consuming and can be complicated when dealing with factors and multiples of large numbers ( 3 digit numbers in particular).

You can therefore use prime factorization to find these values.

The fundamental theorem of arithmetic states that every positive whole number greater than one is either a prime number, or can be written as a product of its prime factors. Every number has a unique set of numbers called prime factors.

By presenting prime factors within a Venn diagram , you can quickly determine both the GCF and LCM of the two or more numbers in the question.

For example,

8=2\times{2}\times{2}

12=2\times{2}\times{3}

The intersection of the two circles contains the greatest common factor , where you multiply the values within the intersection together .

Here, the GCF of 8 and 12 is equal to 2\times{2}=4.

The union of the two circles contains the least common multiple where you multiply the values within both circles together .

Here, the LCM of 8 and 12 is equal to 2\times(2\times{2})\times{3}=24.

As the least common multiple is found by multiplying all of the factors together within the Venn diagram, the least common multiple can be found by multiplying the greatest common factor by the remaining prime factors.

This allows you to solve problems where you are given the GCF and LCM of two numbers and you need to determine the original two numbers.

Common Core State Standards

How does this relate to 6th grade math?

• Grade 6 – The Number System (6.NS.4) Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers (1–100) with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 \, as \, 4 (9 + 2).

[FREE] GCF and LCM Worksheet (Grade 6 to 8)

Use this worksheet to check your grade 6 to 8 students’ understanding of the greatest common factors and lowest common multiples of numbers. 15 questions with answers to identify areas of strength and support!

How to find the greatest common factor

In order to find the greatest common factor of two or more numbers:

State the product of prime factors for each number.

Write all the prime factors for each number into a Venn diagram.

Multiply the prime factors in the intersection to find the GCF.

How to find the least common multiple

In order to find the least common multiple of two or more numbers:

Multiply each prime factor in the Venn diagram to find the LCM.

GCF and LCM examples

Example 1: gcf of two simple composite numbers.

Find the greatest common factor of 30 and 42.

2 Write all the prime factors for each number into a Venn diagram.

3 Multiply the prime factors in the intersection to find the GCF.

GCF =2\times{3}=6.

Example 2: LCM of two simple composite numbers

Calculate the least common multiple of 16 and 18.

16=2\times{2}\times{2}\times{2}

18=2\times{3}\times{3}

LCM =(2\times{2}\times{2})\times{2}\times(3\times{3})=8\times{2}\times{9}=144.

Example 3: GCF word problem

120 \ ml of red paint and 156 \ ml of blue paint are mixed together to create a tin of purple paint. The paint is then distributed equally into sample tubes. Each tube must contain the same amount of paint that must be over 20 \ ml.

What is the maximum number of tubes that can be filled with the minimum amount of paint?

120=2\times{2}\times{2}\times{3}\times{5}

156=2\times{2}\times{3}\times{13}

GCF =2\times{2}\times{3}=12.

The total amount of paint is 120 + 156 = 276 \ ml.

Dividing 276 \ ml into 12 equal shares (the GCF), we have

276\div{12}=23.

As each tube must contain over 20 \ ml of paint, we must have 12 tubes, each containing 23 \ ml of paint.

Example 4: LCM word problem

A plumber is fixing multiple leaking pipes. Pipe A drips water every 12 seconds. Pipe B drips water every 22 seconds. Both pipes drip at the same time. How much time passes before they next drip at the same time? Write your answer using minutes and seconds.

22=2\times{11}

LCM =(2\times{3})\times{2}\times{11}=6\times{2}\times{11}=132

132 seconds pass. Converting this to minutes and seconds is 2 minutes and 12 seconds ( 60 + 60 + 12 = 132 , with 60 seconds = 1 minute).

How to find the original values given the GCF and the LCM

In order to find the original values given the GCF and the LCM:

• Divide the LCM by the GCF.

Calculate the product of primes of the remainder.

Determine which prime factors match each original number.

Example 5: find the numbers, given the GCF

The greatest common factor of 3 numbers is 7. The product of their remaining prime factors is 30 and each number is greater than 10. Determine the value of the three numbers.

Divide the LCM by the GCF to determine the remainder.

As we already know the remainder ( 30 ), we can move on to step 2.

Using a prime factor tree, the product of primes for 30 is:

30=2\times{3}\times{5}

As each value is greater than 10, the GCF 7 must be a factor of all 3 numbers and it must be multiplied by another factor. 30 has 3 prime factors, 2, 3, and 5 and so the original three numbers are:

A=7\times{2}=14

B=7\times{3}=21

C=7\times{5}=35

Example 6: find the original numbers given the GCF and LCM

Two numbers, A and B, have the following number properties:

• GCF (A,B) = 7
• LCM (A,B) = 2,310
• A is divisible by 3
• B is an even number
• 100<A<B

Determine the values of A and B.

2310\div{7}=330

Using a prime factor tree, the product of primes for 330 is:

330=2\times{3}\times{5}\times{11}

As A is divisible by 3, two factors of A must be 3 and 7 (the GCF).

As B is even, two factors of B must be 2 and 7 (the GCF).

Writing this up so far, we have

A=3\times{7}\times{x}

B=2\times{7}\times{y}

As 330=2\times{3}\times{5}\times{11}, we have the remaining factors of 5 and 11 to place.

As 100<A<B, both A and B are greater than 100 with A being smaller than B. The only way this is possible is by making x=5 and y=11.

This means that,

A=3\times{7}\times{5}=105

B=2\times{7}\times{11}=154

The solution is A = 105 and B = 154.

Tips for teaching GCF and LCM

• Before introducing GCF and LCM, students should have a strong understanding of factors and multiples, which they would have first learned in 4th grade. Review these terms and practice listing factors and multiples if needed.

Easy mistakes to make

• Finding the GCF instead of the LCM (and vice versa) A very common misconception is mixing up the greatest common factor with the least common multiple. Factors are composite numbers that are split into smaller factors. Multiples are composite numbers that are multiplied to make larger numbers.
• Incorrect evaluation of powers It is possible to write prime factors into a Venn diagram with their associated exponent or power. This only becomes an issue when the powers are not correctly interpreted. We suggest having students write the prime factors without the use of exponents. Take, for example, the numbers 12 and 18. 12=2^{2}\times{3} 18=2\times{3}^{2} Here, 2^{2}=2\times{2}=4 which is correct, however, the same misconception could then be continued to 3^{2}=3\times{2}=6, which is incorrect. Instead, 3^{2}=3\times{3}=9. This will have an impact on the value of the GCF and the LCM.

Related lessons on factors and multiples

• Factors and multiples
• Factor tree
• Least common multiple
• Greatest common factor
• Prime factors
• Factor pairs

Practice GCF and LCM questions

1. Find the GCF of 54 and 60.

GCF (54,60) = 2\times{3}=6

2. Find the LCM of 24 and 32.

LCM (24,32) = 3\times(2\times{2}\times{2})\times(2\times{2})=3\times{8}\times{4}=96

3. Two lengths of ribbon measure 1.2 \ m and 80 \ cm. Each piece of ribbon needs to be cut into the fewest number of pieces of the same length. What is the length of each piece?

GCF (80,120) = 2\times{2}\times{2}\times{5}=40

4. Two runners leave the start line of a 200 \ m track on the whistle. It takes runner A \ 1 minute to run 1 lap of the track and runner B \ 1 minute and 12 seconds. What distance will runner B have traveled when they next cross the start line at the same time?

Converting both lap times to seconds, runner A takes 60 seconds, and runner B takes 72 seconds.

GCF (60,72)=2\times{2}\times{3}=12

LCM (60,72) = 5 \times 12 \times (2 \times 3)=5 \times 12 \times 6=360

360 seconds = 6 minutes

6\div{1.2}=5 laps

5. The greatest common factor of two numbers is 35. The product of the remaining factors is 33. Both original numbers contain three digits. What is the difference between the two original numbers?

Smaller number: 35\times{3}=105

Larger number: 35\times{11}=385

6. Two numbers x and y have the following number properties:

• \text{LCM }(x,y)=96
• \text{GCF }(x,y)=8
• 2<x<y<40

What is the value of x+y?

GCF and LCM questions

1. A farm needs to divide their two fields into equal-sized enclosures for some horses. Field 1 is 240 \ m^2. Field 2 is 160 \ m^2. Each horse must have at least 42 \ m^2.

(a) What is the minimum possible area for each enclosure?

(b) What is the maximum number of horses that can use these two fields?

240=2^4 \times 3 \times 5 \, or \, 240=2 \times 2 \times 2 \times 2 \times 3 \times 5

160=2^5 \times 5 \, or \, 160=2 \times 2 \times 2 \times 2 \times 2 \times 5

GCF (240,160)=80 \ m^2

2+3=5 \, or \, (240+160) \div 80=5

2. Given that 6480=2^4 \times 3^4 \times 5, simplify the ratio 10800:6480.

GCF (10800,6480)=2^4 \times 3^3 \times 5

Remaining factors are 5 (for 10800 ) and 3 (for 6480 ).

3. The least common multiple of x and y is 2^3 \times 3^2 \times 5^2 where x is a square number such that 36<x<225.

(a) Find the exact value of x.

(b) The greatest common factor of x and y is 4. Determine the value of y. Use the Venn diagram below to help you.

x=2^2 \times 5^2 or

GCF and LCM FAQs

Step 1: State the product of prime factors for each number. Step 2: Write all the prime factors for each number into a Venn diagram. Step 3: Multiply the prime factors in the intersection to find the GCF.

Step 1: State the product of prime factors for each number. Step 2: Write all the prime factors for each number into a Venn diagram. Step 3: Multiply each prime factor in the Venn diagram to find the LCM.

To find the GCF, list all prime factors that are common between the two numbers and multiply them together. To find the LCM, multiply the GCF by all the prime factors of both numbers that have not yet been used.

The least common denominator (LCD) is the least common multiple (LCM) of the denominators of two or more fractions.

The next lessons are

• Converting fractions, decimals, and percentages
• Fractions operations

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GCF and LCM Explained w/ 7 Step-by-Step Examples!

// Last Updated: November 9, 2020 - Watch Video //

Do you ever get GCF and LCM confused?

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

It happens, right?

Well, today, we’re going to learn the one method that gets you the answers to both very easily.

Let’s go!

But first let’s review the basic definitions of each.

What Is GCF And LCM

The Greatest Common Factor (also known as GCF ) is the largest number that divides evenly into each number in a given set of numbers.

The Least Common Multiple (also known as LCM ) is the smallest positive multiple that is common to two or more numbers.

Why Do You Need Both Methods

So will there ever be a time when we will need to use both the GCF, Greatest Common Factor and LCM, Least Common Multiple?

Yes, whenever we perform operations with fractions !

For instance, we may need to use the LCM to help us add two fractions , and also the GCF to simplify our result .

Consequently, you will need to know how to use both of these techniques at the same time.

How To Find GCF And LCM

How do you keep them straight and not mix them up?

Great question!

First, whenever you are asked to find both the greatest common factor and the least common multiple, always choose the prime factorization method , or the listing of prime factors, as it will save you time and is the only method that will work consistently.

And secondly, use the last letters of GCF and LCM to find what you need!

Here’s a trick: GC F = F ewer and LC M = More

Remember, when using our prime factorization technique , we choose the fewest common factors for the GCF, and for the LCM, we choose the most of each factor as discussed at Minnesota State University .

Example #1 — Two Numbers

Working a few problems will help to make sense of how this works.

For our first question, let’s find the GCF and find the LCM of two numbers: 12 and 18

Find GCF and LCM of Two Numbers — Example

This means that the GCF of (12 and 18) is 6, and the LCM of (12 and 18) is 36.

Example #2 — Three Numbers

Now let’s work a problem involving three numbers.

Find the GCF and LCM of 15, 18, 24

Find GCF and LCM of Three Numbers — Example

• The GCF of (15, 18, and 24) is 3.
• And the LCM of (15, 18, and 24) is 360.

Using prime factorization and our trick for remembering what factors to choose is a snap!

Closing Thoughts

Now, I would like to point out that the phrase GCF has many synonyms. So, if you ever hear or see one of these alternate phrases, don’t be alarmed. Just know they all mean the same thing – find the greatest positive integer that divides evenly into two or more numbers.

The alternative terminologies for the Greatest Common Factor (GCF) are:

• Highest Common Factor (HCF)
• Greatest Common Divisor (GCD)
• Greatest Common Measure (GCM)
• Highest Common Divisor (HCD)

And while there are no alternate terminologies for Least Common Multiple, you will hear Least Common Multiple (LCM) and Least Common Divisor (LCD) used together quite often. Sometimes, they will be used interchangeably .

The LCM is how we find common multiples of two or more numbers, whereas the LCD is the least common multiple in a fraction’s denominator. So, the LCD is a subset or special case of the LCM. But in all honesty, they require the same math process, so many teachers and students use these two phrases as synonyms.

But, regardless of what the technique is called, the process for finding the greatest common factor and the least common multiple is very straightforward.

Worksheet (PDF) — Hands on Practice

Put that pencil to paper in these easy to follow worksheets — expand your knowledge!

GCF and LCM — Practice Problems GCF and LCM — Step-by-Step Solutions

Video Tutorial — Full Lesson w/ Detailed Examples

Together we will work through various exercises involving two and three numbers to master the techniques of finding the GCF and LCM and never getting them mixed up.

• Introduction to Video: GCF and LCM
• 00:00:26 – How do you find the Greatest Common Factor and the Least Common Multiple?
• 00:01:45 – Find both the GCF and LCM (Examples #1-3)
• 00:14:17 – Determine the GCF and the LCM of three numbers (Examples #4-7)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions

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Word Problems that uses GCF or LCM (Worksheets)

Related Topics & Worksheets: Least Common Multiple More Math Worksheets

Objective: I can find the least common multiple or least common denominator.

Read the lesson on least common multiple if you need to learn how to find the lowest common multiple.

We use the least common multiple when adding or subtracting fractions with unlike denominators. It is then called the least common denominator.

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GCF and LCM Word Problems Worksheets

Greatest Common Factor (GCF) of numbers is the largest number that divides evenly into those numbers. Lowest Common Multiple (LCM) of numbers is the minimum number of which the numbers are factors. Practice GCF and LCM problems which are available in GCF and LCM worksheets. It will help you in solving lcm and gcf word problems which are practical applications in daily life. Use the Educational apps, Educational videos to become perfect in gcf and lcm word problems.

The apps, sample questions, videos, and worksheets listed below will help you learn GCF and LCM.

Sample questions on gcf and lcm word problems, gcf and lcm word problem apps.

GCF and LCD

Math Tool – Prime Factor

GCF & LCM Quiz Master

Educational Videos related to GCF and LCM – word problems

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Related Topics

What is gcf and lcm.

Greatest Common Factor (GCF) A common multiple is a number that is a multiple of two or more numbers. Common multiples of 2 and 3 are 0, 6, 12, 18, … The least common multiple (LCM) of two numbers is the smallest number (excluding zero) that is a multiple of both of the numbers.

What is the difference between LCM and HCF?

The highest common factor is found by multiplying all the factors which appear in both lists: So the HCF of 60 and 72 is 2 × 2 × 3 which is 12. The lowest common multiple is found by multiplying all the factors which appear in either list: So the LCM of 60 and 72 is 2 × 2 × 2 × 3 × 3 × 5 which is 360.

How do you find the LCM of a number?

For example, for LCM (12,30) we find: Prime factorization of 12 = 2 * 2 * 3 = 22 * 31 * 5. 0 Prime factorization of 30 = 2 * 3 * 5 = 21 * 31 * 5. 1 Using the set of prime numbers from each set with the highest exponent value we take 22 * 31 * 51 = 60. Therefore LCM (12,30) = 60.

What is the difference between the least common multiple and the greatest common factor?

A common multiple is a number that is a multiple of two or more numbers. Common multiples of 2 and 3 are 0, 6, 12, 18, … The least common multiple (LCM) of two numbers is the smallest number (excluding zero) that is a multiple of both of the numbers.

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Gcf and Lcm Word Problems Worksheets

GCF and LCM word problems worksheets can help encourage students to read and think about the questions, rather than simply recognizing a pattern to the solutions. GCF and LCM word problems worksheets come with the answer key and detailed solutions which the students can refer to anytime.

Benefits of GCF and LCM Word Problems Worksheets

GCF and LCM word problems worksheets help kids to improve their speed, accuracy, logical and reasoning skills.

GCF and LCM word problems worksheets give students the opportunity to solve a wide variety of problems helping them to build a robust mathematical foundation. GCF and LCM word problems worksheets help kids to improve their speed, accuracy, logical and reasoning skills in performing simple calculations related to the topic of GCF and LCM.

GCF and LCM word problems worksheets are also helpful for students to prepare for various competitive exams.

These worksheets come with visual simulation for students to see the problems in action, and provides a detailed step-by-step solution for students to understand the process better, and a worksheet properly explained about the GCF and LCM.

Download GCF and LCM Word Problems Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

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• 6th Grade GCF Worksheets

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LCM and GCF

What is the lcm.

"LCM" stands for "least common multiple". Given two numbers, their LCM is the least (that is, the smallest) common (that is, shared) multiple of those two numbers. For instance, given the two numbers 4 and 5 , their LCM is the smallest number that includes each of 4 and 5 as factors; in this example, the LCM is 20 .

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Multiples and Least Common Multiples

What is the GCF?

"GCF" stands for "greatest common factor". Given two numbers, the GCF is the greatest (that is the largest) common (that is, shared) factor of those two numbers. For instance, given the numbers 15 and 18 , their GCF is the biggest number that is a factor of (that is, that divides cleanly into) each of 15 and 18 ; in this example, the GCF is 3 .

How do you find the LCM?

There are two methods for finding the LCM of a pair of numbers. The first method requires that you make lists (very long lists, sometimes) of all the numbers that are multiples of the original numbers; you keep listing until you finally find a match, being some multiple that is common to both listings. The second method requires only the prime factorization of the two numbers, followed by (here's the trick!) making a nice neat grid.

How do you find the GCF?

There are two methods for finding the GCF of a pair of numbers. The first method requires that you make a complete list (a very long list, sometimes) of all the numbers that divide evenly into the original numbers; you then compare the two listings, hoping for a match. If there is a match, this number is a common factor; if there is only the one match, then this is the GCF of the two numbers. If there is more than one match, then you take the largest match as the GCF. If there is no match, then the number 1 is the GCF. (The number 1 is, trivially, a factor of every number.) The second method requires only the prime factorization of the two numbers, followed by (here's the trick again!) making a nice neat grid.

Here's an example that uses the first method:

• Find the GCF and LCM of 2940 and 3150 .

To find the Greatest Common Factor, I need to find all of the factors, prime and otherwise, of each of the two numbers. The best way I've found to do this is to find factor pairs; that is, I'll find one number that divides evenly into the number, and then do the division, which gives me the other factor in the pair. So, grabbing my calculator...

2940: 1×2940, 2×1470, 3×980, 4×735, 5×588, 6×490, 7×420, 10×294, 12×245, 14×210, 15×196, 20×147, 21×140, 28×105, 30×98, 35×84, 42×70, 49×60

3150: 1×3150, 2×1575, 3×1050, 5×630, 6×525, 7×450, 9×350, 10×315, 14×225, 15×210, 18×175, 21×150, 25×126, 30×105, 35×90, 42×75, 45×70, 50×63

(I found these factor pairs by dividing the original numbers in my calculator by progressively larger values, starting at 2 and continuing until the answer after division was smaller than what I'd divided by.)

Now I list the factors in order, from least to greatest:

2940: 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 49, 60, 70, 84, 89, 105, 140, 147, 196, 210, 245, 294, 420, 490, 588, 735, 980, 1470, 2940

3150: 1, 2, 3, 5, 6, 7, 9, 10, 14, 15, 18, 21, 25, 30, 35, 42, 45, 50, 63, 70, 75, 90, 105, 126, 150, 175, 210, 225, 315, 350, 450, 525, 630, 1050, 1575, 3150

The largest value that is in both lists is 210 , so this is the GCF.

Now I have to start listing the multiples of each of the original numbers, until I find a duplicate:

2940: 2940, 5880, 8820, 11760, 14700, 17640, 20580, 23520, 26460, 29400, 32280, 38220, 41160, 44100, 47040, 49980,...

3150: 3150, 6300, 9450, 12600, 15750, 18900, 22050, 25200, 28350, 31500, 34650, 37800, 40950, 44100,...

While I was making my list of multiples of 3150 , I had to keep extending my list of multiples of 2940 , until I finally found a duplicate. This duplicate, 44100 , is the LCM.

Okay; that was the painful way of finding the LCM and GCF. Here's the other, much easier and faster, way:

First, I need to factor factor each of the numbers they've given me:

(This factorization process is pretty easy: divide by the smallest prime that goes in evenly, working your way through the primes until the answer to your division is itself a prime. Review this lesson if you need a refresher.)

Now I'll apply the same sequential-division process to 3150 :

The factorizations can be read from the numbers along the outside of the sequential divisions, so my prime factorizations are:

2940 = 2 × 2 × 3 × 5 × 7 × 7

3150 = 2 × 3 × 3 × 5 × 5 × 7

I will write these factors out, all nice and neat, with the factors lined up according to occurrance:

This orderly listing, with each factor having its own column, will be doing most of the work for me.

The Greatest Common Factor, the GCF, is the biggest (that is, the "greatest") number that will divide into (that is, the largest number that is a factor of) both 2940 and 3150 . In other words, it's the number that contains all the factors *common* to both numbers. In this case, the GCF is the product of all the factors that 2940 and 3150 share.

So, to find the GCF, I just take all the factors that are in both factorizations:

Then the GCF is 2 × 3 × 5 × 7 = 210

On the other hand, the Least Common Multiple, the LCM, is the smallest (that is, the "least") number that both 2940 and 3150 will divide into. That is, it is the smallest number that contains both 2940 and 3150 as factors, the smallest number that is a *multiple* that is common to both these values. Therefore, it will be the smallest number that contains every factor in these two numbers.

So, to find the LCM, I just take all the factors from each of the columns in my factor grid:

Note that I took each column's factor; I did not use all the copies of a given factor. For instance, while 2940 has two copies of the factor 2 and 3150 has one, I did not take *three* copies of the factor 2 . There are only two columns with 2 as their factor, so I took only two copies of that factor when finding the LCM.

Then the LCM is 2 × 2 × 3 × 3 × 5 × 5 × 7 × 7 = 44,100

Many students get confused and accidentally over-duplicate their factors, so let's spend a little extra time on this. Consider two smaller numbers, 4 and 8 , and their LCM. The number 4 factors as 2 × 2 ; 8 factors as 2 × 2 × 2 . The LCM needs only have three copies of 2 , in order to be divisible by both 4 and 8 . That is, the LCM is 8 . You do not need to take the three copies of 2 from the 8 , and then throw in two extra copies from the 4 . This would give you 32 . While 32 is a common multiple, because 4 and 8 both divide evenly into 32 , 32 is not the LEAST (that is, it is not the smallest) common multiple, because you'd have over-duplicated the 2 s when you threw in the extra copies from the 4 .

To reiterate: For the GCF, you carry down only those factors that all of the factor listings share (that is, only those factors whose columns are filled in each row); for the LCM, you carry down all the factors, regardless of how many or few of the original values contained that factor in their listings.

Let the nice neat listing keep track of things for you, especially when the numbers get big.

• Find the LCM and GCF of 27 , 90 , and 84

First, I need to find the prime factorizations:

Then I will list these factorizations neatly, with each copy of each factor getting its own column:

Then the GCF (being the product of the shared factors — so its the product of all the full columns) and the LCM (being the product of all factors — so its the product of all of the columns) are given by:

Then my answer is:

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By the way, if you prefer (or if you're lazy, like me), you can omit the "times" signs in your tables, and just list the factors. It'll look like this:

• Find the GCF and LCM of 3 , 6 , and 8

First I factor the numbers and list their prime factorizations:

Then my GCF and LCM are given by:

Note that 3 , 6 , and 8 share no common factors. While 3 and 6 share a factor, and 6 and 8 share a factor, there is no prime factor that all three of them share. Since 1 divides into everything, then the greatest common factor in this case is just 1 . When 1 is the GCF, the numbers are said to be "relatively" prime; that is, they are prime, relative to each other, because they have no common factor (other than the "trivial" factor of 1 ).

You can use the Mathway widget below to practice finding the LCM or GCF. Try the entered exercise, or type in your own exercise. Then click the button and select "Find the LCM" from the options, and then compare your answer to Mathway's. (Or jump down to the continuation of this lesson.)

Please accept "preferences" cookies in order to enable this widget.

(Click " Tap to view steps " to be taken directly to the Mathway site, if you'd like to check out their software or get further info.)

The GCF doesn't come up that much at this stage in mathematics, though some books use it for factoring polynomial expressions by having the student find the GCF of all the terms in the polynomial and divide this value out of every term. But the LCM comes up every time you need to find a lowest common denominator for fractions.

The factor technique I demonstrated above works even for polynomial fractions. (The other method for finding the LCM, the "listing" method, will not work for polynomials, which is why you will need to learn the factor method eventually.) If you need to find the LCM of two (or more) polynomials, you can do the exact same procedure as above:

• Find the LCM of x 3  + 5 x 2  + 6 x and 2 x 3  + 4 x 2

First I factor the polynomials:

x 3 + 5 x 2 + 6 x

x ( x 2 + 5 x + 6)

x ( x + 2)( x + 3)

2 x 3 + 4 x 2

2 x 2 ( x + 2)

Then I list these factors out, nice and neat:

The LCM is the product of the entries from all of the columns:

I take two copies of " x ", because 2 x 3 + 4 x 2 contains two copies. I don't need three copies of " x ", because neither polynomial contains three copies. I need only one copy of x + 2 , because neither polynomial contains more than just the one copy. I need to account for the 2 from the second polynomial and the x + 3 from the first polynomial.

LCM: 2 x 2 ( x  + 2)( x  + 3) .

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Finding GCF and LCM with the Ladder (or Cake) Method

7 comments:

The cake method is a cool way, however it could become problematic if we try to use it to find LCM for 3 (or more) numbers. Example: LCM of 8, 12, 16 - the cake method will show GCF of 4 with remaining 2, 3, 4 and student will think the LCM is 4x2x3x4=96, but in reality the LCM is 48. Because of that, the prime factorization/factor tree might be safer method. But the cake method will be good if only work with 2 numbers for GCF & LCM. :D

Yes I came to say the same thing. I only use the cake method for GCF because it works when you need the GCF of 3 numbers but for LCM they need an alternative method anyway so I stick with listing the multiples on the M.

UNLESS you're like me. I just found out the proper way to use the CAKE METHOD. If you are using 3 numbers-it still works. For example, the LCM of 8,12, 16. I started with dividing them all by 4 which gave me the remainders of 2, 3, 4. YOU CAN DIVIDE ONLY TWO OF THE THREE-WHICH MEANS I CAN DIVIDE THE 2 AND 4, NOT THE 3. IF YOU CAN'T DIVIDE IT, YOU JUST BRING IT DOWN AS A REMAINDER. MIND BLOWN. SO once you do that, you end up with 1, 3 and 2 in your final layer. THEN making the shape of an L would give you 4x2x1x3x2=48. IT DOES WORK EVERYTIME! Hope this helps!

It works if you do it in multiple steps - first using 8 and 12 to get an LCM of 24. Then using 24 and 16 to get an LCM of 48. Or you could start with 8 and 16 (16) and then do 16 and 12 to get 48. Or start with 12 and 16 (48) and then do 12 and 48 (still 48).

Alternatively, you can do all 3 numbers at once first factoring out everything possible from all 3 numbers, then factoring out anything from 2 numbers and just dropping down the number you didn't divide. In this case after dividing 4 you then chose 2 and get 1, 3, 2 remaining on the bottom. The LCM is 4*2*1*3*2=48 which is the actual LCM. You stop at 4 for the GCF though, that's only the things taken out of all 3 numbers multiplied together. I love this method, it's so amazing!

My concern with this method is that it doesn't move them toward algebra readiness. We switch to the prime factorization method so they are ready to apply it to finsing the GCF and LCM of polynomials in their study of rational expressions and equations. Are we going to start teaching the cake method for polynomials also? Otherwise with this easy pass right now, we're setting them up for failure later on.

Valid concern! The cake method can be used for polynomials, too. I think there's value in showing students that there are multiple ways to approach math problems and that part of their job as students is to find the method that works best for them :)

Home / United States / Math Classes / 6th Grade Math / GCF and LCM

GCF and LCM

In mathematics, a factor of a natural number is the number that divides a number and results in zero as a remainder. On the other hand, the multiple of a natural number is a number that is obtained by multiplying a given number with another natural number. In this article we will learn about GCF (HCF) and LCM based on factors and multiples. ...Read More Read Less

About GCF and LCM

What is GCF?

Methods to find gcf(hcf).

• Listing of Factors
• Prime Factorization

What is LCM?

Methods to find lcm.

• Listing Multiples

Solved Examples

• Frequently Asked Questions

The GCF stands for greatest common factor , it is also known as highest common factor(HCF). GCF is the greatest among the common factors of a number. The factors that are shared by two or more numbers called common factors. 

In this method, we list the factors of two or more numbers and find the common factors from the list. Then we select the greatest among all the common factors.

In this method, we use a factor tree to find the prime factorization of numbers and list out the common prime factors of the numbers. Then the product of all the common prime factors results in the greatest common factor.

The LCM stands for least common multiple . The LCM is the least multiple among all the common multiples of two or more numbers.

As we already know, a multiple is the number obtained by multiplying a number by itself or any other number.

In this method we list the multiples of two or more numbers and  find out the common multiples from the list. Then, the least among all common multiples is the least common multiple.

In this method we use the factor tree to find the prime factorization of the numbers. List each different factor where it appears the greatest number of times and multiply them to get the least common multiple.

Example 1. Find the greatest common factor of 24 and 30 using listing the factors method.

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.

Circle the common factors.

The common factors of 24 and 30 are 1, 2, 3, and 6. Hence, the greatest common factor of 24 and 30 is 6.

Example 2. Find the greatest common factor of 28 and 44 using prime factorization.

Use factor tree method to get prime factors of numbers:

28 = 2 x 2 x 7

44 = 2 x 2 x 11

So, the GCF of 28 and 44 is 2 x 2 = 4.

Example 3: Find the least common multiple of 4 and 5. Use the listing of multiples method.  

Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, …

Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, …

The common multiples have been encircled below:

The common multiples of 4 and 5 are 20, 40, and so on. The least of these common multiples is 20.

Hence the LCM of 4 and 5 is 20.

Example 4: Sam writes a poem every 2 days and Annie writes a poem every 3 days. Today both Annie and Sam are writing poems. After how many days do they write a poem together again?

You are given the number of days Sam and Annie take to write a poem. The LCM of the number of days will give us the time when they will write together again from this day. So, let’s find the LCM of 2 and 3.

Multiples of 2 are 2, 4, 6 , 8, 10, 12 , 14, 16, 18 , 20.

Multiples of 3 are 3, 6 , 9, 12 , 15, 18 , 21, 24, 27, 30.

The common multiples of 2 and 3 are 6,12 and 18. The least of these common factor is 6

so, the LCM of 2 and 3 is 6.

Hence, Sam and Annie will write together again on the 6th day from the current day.

What is meant by GCF?

GCF or the greatest common factor is the greatest of all common factors between any two or more numbers. 

What is meant by LCM?

LCM or the least common multiple is the least among the common multiple of two or more numbers.

What is the difference between GCF and LCM?

GCF is the greatest number that is a common factor of two or more numbers. However, LCM is the least among the common multiple of two or more numbers.

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GCF and LCM Calculator

How to use the gcf finder.

The GCF and LCM calculator (also called the GCF finder) will determine the greatest common factor and least common multiple of a set of two to six numbers . You can also compute the GCF and LCM by hand or use the GCF calculator or the LCM calculator to find more detailed methods to compute these problems by hand.

If you want to find the GCF and LCM, first, you need to get the prime factorization of each number in the set. This is done easily with the prime factorization calculator .

Suppose you want to find the GCF and LCM of 24 and 56 .

First we get the prime factorizations of 24 = 2 × 2 × 2 × 3 and 56 = 2 × 2 × 2 × 7 .

The greatest common factor is what is present in both sets of factors, which is 2 × 2 × 2 = 8 .

The least common multiple is the highest power of all exponents, which is 2 × 2 × 2 × 3 × 7 = 168 .

There are several methods for finding GCF, including prime factorization or the Euclidean algorithm using the modulo calculator . The factor calculator is also a handy tool for finding GCF and LCM. Note that while finding the GCF and LCM of smaller numbers is relatively simple by hand, the GCF and LCM calculator is quicker and much easier for larger or larger sets of numbers.

What is the GCF?

The GCF , or greatest common factor , is the highest number that divides exactly two or more numbers. For example, the greatest common factor of 20 and 16 is 4 , as both numbers can be divided by that value: 20/4 = 5 , 16/4 = 4 .

How do I calculate the GCF?

To find the greatest common factor in any set of numbers, follow these easy steps:

• Write the prime factorization of the numbers.
• Select all the factors shared by the factorizations , with the highest exponent .
• Multiply the shared factors.

That's it! The hardest part of this process is finding the prime factors; the rest is straightforward.

What is the GCF of 8, 36, and 12?

The GCF of 8 , 36 , and 12 is 4 . To find it:

Write the prime factors of the three numbers:

8 = 2 × 2 × 2 = 2³ ;

36 = 2 × 2 × 3 × 3 = 2² × 3³ ; and

12 = 2 × 2 × 3 = 2² × 3 .

Find the factors that repeat in both factorizations. In this case, we have only 2² .

4 is the greatest common factor as:

• 36/4 = 9 ; and

What is the least common multiple?

The least common multiple of a set of numbers is the smallest number greater than each value in the set that is exactly divisible by all numbers in the set . To find the least common multiple, follow these steps:

Write the prime factorizations of the numbers in the set.

Identify all the factors , and chose the highest power in which they appear .

Multiply the factors (and their powers, in case) to find the least common multiple.

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