how to solve gcf and lcm word problems

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.

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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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LCM Questions

LCM Questions are given here, along with detailed solutions and proper explanations to help out students regarding the concept of LCM. These questions on LCM will help the students to be able to solve the problems efficiently. Learn more about What is LCM?

The LCM or Lowest Common Multiple of two or more numbers is the least among all the common multiples of given numbers. For example, LCM of 2, 4 and 5 is 20, which is the lowest common multiple of 2, 4 and 5, or we can say 20 is the lowest number which 2, 4 and 5 can divide.

LCM Questions with Solutions

1. If HCF(252, 594) = 18, find LCM(252, 594).

Solution: We have LCM of two numbers = (Product of two numbers)/ their HCF

= (252 × 594)/18 = 8316.

Hence, LCM(252, 594) = 8316

2. Player 1 and player 2 are running around a circular field. Player 1 takes 16 minutes to take one round, while Player 2 completes the round in 20 minutes. If both start simultaneously and go in the same direction, after how much time will they meet at the starting point?

Solution: Time taken by the players to meet again = LCM(16, 20)

Now 16 = 2 4 and 20 = 2 2 × 5

Therefore, LCM(16, 20) = 2 4 × 4 = 80

Hence, both will meet at the starting point after 80 minutes.

3. Is it possible to have two numbers whose HCF is 18 and LCM is 540?

Solution: Since HCF always divides LCM, we see that 540 is divisible by 18, 540/18 = 30.

Thus, it is possible to have two numbers.

4. Find the least number divided by 28 and 32, leaving the remainder 8 and 12, respectively.

Solution: Since 28 – 8 = 20 and 32 – 12 = 20

So we need LCM{(28, 32) – 20} = 224 – 20 = 204

Thus, 204 is required number.

5. Find the least number, which, when divided by 35, 56 and 91, leaves the same remainder of 7, respectively.

Solution: Let us find LCM(35, 56, 91) + 7

56 = 2 × 2 × 2 × 7

91 = 7 × 13

Thus, LCM(35, 56, 91) = 2 3 × 5 × 7 × 13 = 3640

The required number = LCM(35, 56, 91) + 7 = 3640 + 7 = 3647.

6. Two alarm clocks ring their alarms at regular intervals of 72 seconds and 50 seconds. If they beep together at noon, at what time will they beep again for the first time?

Solution: We find the LCM of 72 and 50.

Prime factorisation of 72 and 50,

72 = 2 × 2 × 2 × 3 × 3

50 = 2 × 5 × 5

Therefore, the LCM of 72 and 50 = 2 3 × 3 2 × 5 2 = 1800

1800 seconds = 1800/60 min = 30 min

Hence, the clocks will beep again for the first time at 12:30 pm.

7. There are 56 students in section A and 58 students in section B of a class in a school. Find the minimum number of books required for their class library so that they can be distributed equally among the students of section A or section B.

Solution: Clearly, the number of books that are to be equally distributed should be multiple of 56 and of 58. Thus, we have to find LCM of 56 and 58.

Now, 56 = 2 × 2 × 2 × 7

58 = 2 × 29

LCM (56, 58) = 2 3 × 7 × 29 = 1624.

Hence, atleast 1624 books are required in the library.

• Euclid’s Division Lemma
• Real Numbers
• Rational and Irrational Numbers
• Polynomials

8. Find the LCM of 96(x – 1)(x + 1) 2 (x + 3) 3 and 64(x 2 – 1)(x + 3)(x + 2) 2 .

Solution: Let f(x) = 96(x – 1)(x + 1) 2 (x + 3) 3

And g(x) = 64(x 2 – 1)(x + 3)(x + 2) 2

Now, factorising the polynomials into irreducible factors.

f(x) = 2 × 2 × 2 × 2 × 2 × 3 × (x – 1)(x + 1) 2 (x + 3) 3

g(x) = 2 × 2 × 2 × 2 × 2 × 2 × (x + 1)(x – 1)(x + 3)(x + 2) 2

Taking all the factors raised to their highest exponents: 2 6 , 3, (x – 1), (x + 1) 2 , (x + 3) 3 , (x + 2) 2

⇒ The LCM of the given polynomials = 192(x – 1)(x + 1) 2 (x + 2) 2 (x + 3) 3 .

9. Find the LCM of ⅔, ¾ and 7/2.

Solution: LCM of ⅔, ¾ and 7/2 = [LCM of (2, 3, 7)]/[HCF of (3, 4, 2)]

LCM of (2, 3, 7) = 2 × 3 × 7 = 42

HCF of (3, 4, 2) = 1

Therefore, LCM of ⅔, ¾ and 7/2 = 42.

10. Find the LCM of 22.5, 3.5 and 0.55.

Solution: Converting the decimals into integers,

22.5 = 22.5 × 100 = 2250

3.5 = 3.5 × 100 = 350

0.55 × 100 = 55

Now, 2250 = 2 × 3 × 3 × 5 × 5 × 5

350 = 2 × 5 × 5 × 7

55 = 5 × 11

LCM (225, 350, 55) = 2 × 3 2 × 5 3 × 7 × 11 = 173250

Place a decimal point after place from right

Then, LCM(22.5, 3.5, 0.55) = 1732.5.

11. Find the smallest number, which is, when reduced by 7, is divisible by 12, 16, 18, 21 and 28.

Solution: Let x be a number when reduced by 7, is divisible by 12, 16, 18, 21 and 28.

Then, x – 7 = m × 12 ⇒ x = m × 12 + 7.

Thus, when x is divided by 12, it leaves a remainder of 7, which is the same for each given number.

∴ x = LCM (12, 16, 18, 21, 28) + 7 = 1008 + 7 = 1015.

12. Find the largest four-digit number, which is divided by 4, 7 and 13, leaving a remainder of 3, respectively.

Solution: Largest 4-digit number = 9999

LCM (4, 7, 13) = 364

Now 9999 = 27 × 364 + 171

Thus, the 4-digit number divisible by 4, 7 and 13 = 9999 – 171 = 9828.

Since the number we have to find leaves a remainder of three when divided by 4, 7 and 13,

∴ 9828 + 3 = 9831 is the required number.

13. Six bells commence tolling together. After that, they toll at a time interval of 2, 4, 6, 8, 10 and 12 seconds, respectively. In 60 minutes, how many times do they toll together?

Solution: Taking LCM (2, 4, 6, 8, 10, 12) = 120

So, the bells will toll together after each 120 seconds = 2 min

∴ In 60 minutes number of times the bells will toll = 60/2 + 1 = 31 times.

Video Lesson on Application of LCM

Related Articles:

Practice questions:.

1. If the LCM of 12x 3 y 2 and 18x p y 3 is 36x 4 y 3 . Find the value of p.

2. The sum of LCM and HCF of the two numbers is 1260. If their LCM is 900 more than their HCF, find the product of two numbers.

3. The LCM and HCF of two numbers are equal; the numbers must be ______.

4. Two bells toll at an interval of 24 minutes and 36 minutes, respectively. If they tolled together at 9 am, after how many minutes do they toll together again, at the earliest?

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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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• How to find the GCF and the LCM word problems

" class="arrow-title-img"> How to find the GCF and the LCM word problems

This enriching resource on how to find the GCF and LCM word problems is here to improve your kids' technical skills and performance in solving word problems efficiently.

Also, we will include effective problem-solving and instructional working strategies so that learners can easily tackle word problems concerning GCF and LCM. With this, your kids can wave goodbye to all the substantial challenges they face or encounter in solving math word problems.

In fact, with this resource, kids can easily identify the problem type and translate the problem into a solvable mathematical equation, thereby providing an accurate result for the word problem.

Steps on how to find the GCF and LCM word problems

Your kids will be excited as they discover simple steps on how to find the GCF and LCM word problems . These steps will model and help young learners easily identify the relational statements in any word problem they encounter. We will begin by designing easy rules to find GCF /LCM.

Also, our resource will guide your kids through identifying key problem features in word problems concerning HCF/LCM. And how to transform math problems into meaningful maths equation.

As we know, learning with examples is usually a great lesson for kids. Hence, this resource will present real-life situation examples to demonstrate how these steps work

Step 1: IDENTIFY:

• You have to start by reading the whole word problem very carefully.
• Then, figure out the scenario that the problem needs you to tackle.
• Last but not least, use the keywords found in the word problem to help you identify the operation you are supposed to carry out efficiently.

Speaking of keywords:

• When looking for GCF in a word problem, these are some of the general keywords you'll come across in the word problem: - greatest, largest, maximum, highest, etc.
• Another clue to know if you need to find the GCF in a word problem is when the problem presents itself as "groups," "rows," "sections," "how many sets," "divided into equal groups," "identical" "split," "share equally" "same combination," "same" "even amount," etc.
• When finding the LCM , these are some general keywords that you'll find in the word problem: - least, minimum, smallest, lowest, etc.
• Similarly, other clues to know if you are going to find but the LCM in a word problem is when the problem presents itself as an event that will repeat itself , when you find the word "every…" in the word problem, etc.
• ***One key Element for learners to understand is that they should not always rely on keywords alone . That is to say; the same keyword can have different meanings in different word problems .
• For this reason, we reiterate on the importance of reading the question very carefully to understand the situation that the word problem is describing , then figure out exactly which operation to use***

Step 2: STRATEGIZE:

At this stage, you should ask yourself, "how am I going to solve the problem?"

• You already know that, from the keyword(s) in the word problem, you will know if you need to look for the LCM or GCF .
• However, you don't have to depend entirely on keywords . Always try to understand the situation that the problem is describing very well before you start solving it.
• After knowing which operation you will perform, construct short expressions/sentences representing the given word problem.

Step 3: SET UP:

Now, write down a numerical expression representing the information given in the word problem.

Step 4: PROVIDE A SOLUTION:

From step 3 above,

If the word problem needs you to find the GCF , use any of the methods listed below:

• The factor tree and column method,
• The slide method,
• The prime factor method.

If the word problem needs you to find the LCM , use one of the following methods:

• Listing the multiples method,
• Prime factorization method,
• Division method,
• The factor tree and column method.

Step 5: CHECK YOUR WORK:

Finally, ask yourself this question. "Does my answer make sense?" If "YES," you are done. If "NO," go back to step 1 and start all over again.

You can always do this by solving to prove your answer, as shown in the attached example below.

Find more practices with these tests online

• Greatest common factor word problems
• Lowest common multiple word problems
• number theory

GCF and LCM word problems

Convert between standard and scientic notation

Compare numbers written in scientific notation

Examples on how to solve the GCF and LCM word problems

Example one: how to find the gcf word problem.

Step 1: First, read the problem carefully and identify the keyword(s) found in the word problem. After doing this, you see that the keywords in the problem are "largest" and "each."

Step 2: Next, how will you solve the problem? From the situation the problem describes, the keyword found in the word problem and the clue, which is "must be shared equally," calls for you to look for the GCF .

With this in mind, construct short expressions/sentences to represent the given word problem.

• Number of cans of juice bought= 100
• Number of hamburgers bought = 50
• Number of packets of potato chips = 50
• Therefore, the largest number of students that can go for the field trip for all the food to be shared equally with no leftovers = the GCF of 100 and 50.

Merge Step 3 and 4: when dealing with the GCF, step 3 and 4 are best understood when merged.

• Start by writing down a numerical expression to represent the bolded sentence in step 2 above, then find the GCF of 100 and 50 using the prime factor method.
• Firstly, find the prime factors of each number 50 = 2 × 5 × 5. 100 = 2 × 2 × 5 × 5
• Secondly, find and highlight the prime factors that the factors have in common. 50 = 2 x 5 x 5 . 100 = 2 x 5 x 2 x 5 .
• Finally, to find the GCF, multiply their common factors together. 2 × 5 × 5 = 50

So, 50 students can go for the field trip if all the food must be shared equally with no left over.

GCF of 100 and 50 = 50

• 100 cans of juice ÷ 50 = 2 cans of juice.
• 50 hamburgers ÷ 50 = 1 hamburger
• 50 packets of potato chips ÷ 50 = 1 packet of potato chips

So, each student will get 2 cans of juice, 1 hamburger, and 1 packet of potato chips.

Step 5: Finally, check your work to know if your answer makes sense. You can do this as shown below.

To check if your answer is correct,

Firstly add the number of cans of juice, hamburgers, and packets of potato chips that each student will get = 2 + 1 + 1 = 4

Secondly, add the total number of cans of juice, hamburgers, and packets of potato chips that they bought = 100 + 50 + 50 = 200

Finally, divide the total number of cans of juice, hamburgers, and packets of potato chips by the total number of cans of juice, hamburgers, and packets of potato chips that each student will get = 200 ÷ 4 = 50

So, since the final answer equals the GCF of 100 and 50, it implies that your answer is correct and makes sense.

Example two: how to find the LCM word problem

Step 1: After carefully reading the problem, you see that the keyword in the word problem is "least."

Step 2: Next, how will you solve the problem? As you can see, the problem described and the keyword found in the word problem show that we should find the LCM .

With this in mind, form short expressions/sentences to represent the given word problem.

• Packages of sandwich rolls = 12
• Packages of sausages = 6
• Packages of hamburger buns = 8
• Packages of hamburger meat patties = 6
• Therefore, the least number of people she can serve if she buys the least number of packages of each item = the LCM of 12, 8, and 6.

Merge Step 3 and 4: Fwhen dealing with LCM, step 3 and 4 are best understood when merged.

*Now, write down a numerical expression to represent the bolded sentence in step 2 above and then proceed to find the LCM of 15 and 40 using the listing multiples method.

• Firstly, list and find the multiples of 12, 8, and 6. 12 = 12, 24, 36 … 6 = 6, 12, 24, 30, 36 … 8 = 8, 16, 24, 36 …
• Secondly, find and highlight the common multiples. 12 = 12, 24 , 36 … 6 = 6, 12, 24 , 30, 36 … 8 = 8, 16, 24 , 36 …
• Thirdly, the common multiples of the numbers are 24 , 36 …
• Finally, the least common multiple of the numbers is 24 .

So, she can serve 24 people if she buys the least number of packages of each item.

To check if your answer is correct, divide the least common multiple of 12, 8, and 6 by the number of items they bought, i.e., 4 items.

Result: 24 ÷ 4 = 6

Then if you multiply the result by 4 (the number of items) and it gives you the least multiple of 12, 8, and 6, it implies that your answer is correct.

6 × 4 = 24

So, since the final answer is equal to the LCM of 12, 8, and 6, it implies that your answer is correct and makes sense .

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Calcworkshop

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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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.

☛ Check Grade wise GCF and LCM Worksheets

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Tags: Least common multiple word problems , Factors and multiples word problems , 7th Grade GCF practice problems , Grade 7 examples of GCF word problems , GCF and LCM examples

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How To Solve GCF Word Problems

Helping students turn a problem about GCF or LCM into a concrete model will help them understand what they are looking for and what steps they need to take to find the solution. The visual model helps them make sense of the situation so that they are not just relying on memorized steps.

A classic GCF problem has two quantities that need to be divided evenly into the greatest number of groups.  

Mr. Gomez has 12 stickers and 18 gel pens. He wants to make the greatest number of packs possible with an equal number of stickers and an equal number of gel pens in each pack. How many packs can he make? 

At this point, many students might confuse this with a LCM problem and say the answer is 36. Others will use a common factor such as 2 or 3 for their answer.

• How can students use a model to solve GCF word problems?

First attempt

Second attempt, third attempt, why should students use a model to solve gcf word problems, frequently asked questions on solving gcf word problems, 1. what is gcf, 2. how can students use a model to solve gcf word problems, 3. why should students use a model to solve gcf word problems, 4. where can i find more worksheets on solving gcf word problems .

100+ Free Math Worksheets, Practice Tests & Quizzes

GCF problems are all about equal groups. In order for a number to be a solution, we must be able to divide both numbers evenly into that number of groups.

It is good for students to act this out or draw this. Here we show you how students can make a drawing to model this situation. Notice the use of color in this model to represent stickers and pens.

Students can put all the stickers and pens into one bag but of course we can try making a few more bags – more fair to have more students getting fewer items than all goodies going to one student!! 

The model shows that we could have 2 bags or 3 bags, because 12 stickers and 18 gel pens can be divided so that there would be the same number in each bag.

But is this the greatest number of bags? Four bags will not work because 18 cannot be divided into 4 equal groups.

The strategy can continue until all possible groups have been tested. The solution is the greatest number of groups that work for both numbers. The model also provides an opportunity to describe the answer. There will be 6 bags. Each bag will have 2 stickers and 3 gel pens.

Drawing the groups can be time-consuming for larger numbers, but using counters works well too. Creating the arrays helps deepen the connection between factors, using arrays for multiplication, and area models.

By using a model, students have moved away from blindly applying GCF or LCM. They make sense of the situation. They try out different options. They don’t have to remember or guess what the concept is supposed to be. 

We could give different pairs of students different situations, ask them to do a poster and then conduct a gallery walk. Students will notice that all factors work and that the greatest common factor is what ends up solving the problem. It also helps them to see what actually is happening inside of the groups – how many stickers and pens are in each group.    

Also read: How To Solve LCM Problems

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It is the largest of all the factors that are common to two or more numbers. Example: GCF of 12 and 18. You can list all the factors of 12 and 18. Then find the greatest common factor. 12: 1, 2, 3, 4, 6 , 12 18: 1, 2, 3, 6 , 9, 18 GCF(12, 18) = 6

When using the model to solve GCF word problems, all the common factors like 1, 2, 3, 6 represent the number of groups that can be formed. 6 represents the largest number of groups that can be formed.

Students should create a chart with two columns and multiple rows. The two columns stand for the items being split up. Each row represents the number of groups the item is split into. Students keep trying more and more groups till they have exhausted all options. 

Students can work in pairs in different scenarios. Conduct a gallery walk with student work. Students will soon come to the conclusion that only common factors work for a number of groups and that the GCF is the largest number of groups you can make.

Solving GCF and LCM word problems is confusing. By using a model, students make sense of the situation. They do not need to memorize the steps. 

You can find math worksheets on GCF problems and other exclusive math resources, here .

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HCF and LCM word Problems, tricks & Solved examples

• The product of the two numbers is always equal to the product of their HCF and LCM.
• In case of HCF, if some remainders are given, then firstly those remainders are subtracted from the numbers given and then their HCF is calculated.
• In case of LCM, if a single remainder is given, then firstly the LCM is calculated and then that single reminder is added in that.
• In case of LCM, if for different numbers different remainders are given, then the difference between the number and its respective remainder will be equal. In that case, firstly the LCM is calculated, then that common difference between the number and its respective remainder is subtracted from that.
• Sometimes in case of HCF questions, the required remainder is given and when the remainder is not given, in those cases you will generally have three numbers given. For answering the question, you need to take the difference of the three pairs of numbers, now the HCF of these differences will become the answer e.g. if you have to find the greatest number, which when divides 83, 93 and 113 and leaves the same remainder. Here you will take the three differences i.e. 93 – 83 = 10 ; 113 – 93 = 20 ; 113 – 83 = 30, after that find the HCF of these differences, which comes out to be 10. Now you can check for yourself- when 10 divides these three numbers, the reminder obtained is 3 in each case and that is what the question was asking for.
• Whenever the question talks about the greatest or maximum, then in most of these cases it will be a question of HCF. Secondly, whenever the question is related to classification or distribution into groups, then in all the cases it is HCF only.
• Whenever the question talks about the smallest or minimum, then in most of the cases it will be a question of LCM. Secondly, whenever the word ‘together’ or ‘simultaneous’ is used in the question, then in all the cases it is LCM.
• Before solving the problems on HCF and LCM in the real exam, you must practice some HCF and LCM worksheets.
• HCF and LCM Concepts
• Problems, tricks & Solved examples
• HCF and LCM Problems (Basic)
• HCF and LCM Problems (Advanced)

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