Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator to add two fractions or to find the reduced form for a fraction).

Find the LCM, least common multiple, of two or more numbers. Use common multiples to find equivalent fractions. This is necessary to find common denominators to create like fractions for the purpose of adding or subtracting. Find the greatest common divisor of the two terms of a fraction and use them to reduce fractions to simplest form.

Find the greatest common factor (GCF) of 12 and 36.

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Jillian was solving the math problem, 2 7/8 + 1 ½. She began by changing 1 ½ to 1 4/8. What steps did she take to make this conversion? Why was this step necessary?

(She found the LCM and used it as the denominator for both fractions. The LCM of 2 and 8 is 8. To change 1 ½ to 1 4/8 she changed the denominator to an 8 by multiplying 2 x 4 = 8. Then, she multiplied the numerator by the same number (1 x 4 = 4). Her new fraction is 4/8. She needed to do this, because in adding fractions or mixed numbers, you must have a common denominator.)

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Stella solved the problem:

Solve 2/3 x 4/6 and write your answer in simplest form.

Her answer was 8/18. Is this the correct answer? Explain.

(Stella’s answer is incorrect, because it is not in simplest form. Both 8 and 18 are divisible by 2.)

Children often confuse the terms greatest common factor and least common multiple. Have your child say aloud the words multiple and multiply. They are in the same word family, because they share the word root multi-. Have him/her think about what the relationship might be between the words. Ask what they might have in common? Tell him/her that to find a number’s multiples, you must always multiply. For example, some of the multiples of 3 are 3, 6, 9… This is because 3 x 1 = 3, 3 x 2 = 6, 3 x 3 =9. Thus, to find the LCM, least common multiple of two numbers you need to think of, or even list all the multiples of them until you find the lowest number they both have in common. An easy way to list the multiples is shown in the sample below. Sample: Find the LCM of 3 and 4.

3: 3, 6, 9, 12

4: 4, 8, 12

The least common multiple of 3,4 is 12.

It is best for children to start out writing a list so that they can visually see the multiples, but as they get familiar with the concept they should be able to compute the LCM mentally.

To find the greatest common factor you can do the opposite as you would with to find the LCM, divide! The purpose of dividing would be to find the factors that make up your number. (see below)

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Remind your child that many numbers have more than one common multiple. It will save him/her time in working with fractions if he/she uses the least common multiply. If you don’t use the LCM, you’ll have to reduce the fraction to lowest terms later. For example, when solving the problem 1/6 + 2/ 3, the fractions need to be changes to like fractions. Many students will change both denominators to 12 because it is a multiple of both 6 and 3. That would mean the like fractions would be 2/12 + 8/12. The sum of these two fractions is 10/12. To place this sum in simplest terms, you’d need to divided by the greatest common divisor. However, this step could’ve been skipped if the least common multiple had been used. The LCM of 1/6 + 2/3 is actually 6, since this is the least common multiple to both denominators. Using the LCM, the like fractions become 1/6 + 4/6. The sum of these two fractions is 5/6. This fraction is in simplest form and does not require any extra steps.

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The math terms greatest common factor and greatest common divisor are both used to define the same math concept. The word divisor, simply implies that you will use the greatest common factor in a division problem. (You may want to point out to your child that the words divisor and divide are in the same word family. This will once again clarify the difference between greatest common divisor and least common multiple.) The term greatest common divisor (GCD) is used in reducing fractions, due to the fact that you will need to divide both terms of a fraction by a common divisor in order to reduce it to simplest terms. Here’s an example of using the GCD to reduce the fraction 9/15. Think of the largest number that can go into both 9 and 15. To do this, first we need to think of the factors that make a 9: 1 x 9 = 9, 3 x 3 = 9. This means the factors of 9 are 1, 3, and 9. Now, we do the same for 15 (1 x 15 = 15, 3 x 5 = 15). The factors for 15 are 1, 3, 15. So, the greatest common factor is 3, because both terms have a factor of 3. Next, we will use 3 as a divisor to reduce the fraction 9/15 (93=3, 153=5). This means the reduced form of 9/15 is 3/5.

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If your child is having difficulty finding the greatest common divisor mentally, he/she may need to take extra steps to show work for a while. One way to find the greatest common divisor of the fraction 12/16 is to use a factor tree. Here’s an example of finding the GCD of 12 and 15:

16

3 4 2 8

2 22 4

2

*The factors of 12 are: 2 x 2 x 3

*The factors of 16 are: 2 x 2 x 2 x 2

Be sure to include all the prime numbers listed on the tree.

Remember, we’re looking for the greatest common divisor. As you can see by looking at both lists of factors. 12 and 16 both have two 2s in common. (They both have 2 x 2). This mean to find the greatest common factor you just multiply the two terms together. So, the greatest factor they both have in common is 4. This means that 4 can be used as a divisor to reduce 12/16 to lowest terms (124=3 and 164=4, so 12/16 = ¾). The simplified form of 12/16 is ¾.

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For the visual/kinesthetic approach use a multiplication chart. Here’s how to use it to find an LCM. First, have your child put his/her fingers on the two numbers that need to be compared. Next, have him/her move their fingers across the chart until they see the first number that appears in both rows. For example, if you’re trying to find the LCD of 4 and 7. Move one finger across the row that shows the multiple of 4. (You may even want to have your child read the multiples aloud). Now, move your other finger across the 7. You will see that the first number that both rows have is 28 This means the LCM of 4 and 7 is 28. You can use the chart similarly, working backward to find the GCF.

What is the greatest common factor of 4, 10, and 20?

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Find the LCM of 12 and 15.

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What is the least common multiple of 4, 10, and 20?

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What is the greatest common divisor (GCD) of the terms of the fraction 12/22?

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What is the GCD of the terms of the fraction 20/140?

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Explain how to use the GCD of 32/48 to put the fraction in lowest terms.

(divide both terms (numerator and denominator) by 16 to get 2/3)

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One multiple of the number 4 is 20. knowing this, how can you write an equivalent fraction for ¾, using 20 as the denominator?

(3/4 = 15/20, since 4 x 5 = 20, you need to multiply the numerator by 5. 3 x 5 = 15.

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Choose a multiple of 6 to write an equivalent fraction for 4/6.

(ex. 8/12)

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What is the greatest common divisor of the fraction 24/96?

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Find the greatest common factor of 40 and 50.

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Find the greatest common factor of 8 and 42.

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Find the greatest common factor of 11 and 13.

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Find the greatest common factor of 123 and 18.

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Find the least common multiple of 32 and 16.

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If Juan is solving the math problem, 6/5 – 1/10, what does he need to do before he can begin subtracting?

(He needs to make the denominators like. Since 10 is the LCM of 5 and 10, this will be the new denominator. He will need to change 6/5 into 12/10.)

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Aidan needs to solve the addition problem 2-1/3 + 4/5. He knows that the LCM of 3 and 5 is 15. How will he use 15 to rewrite the problem before it can be solved?

(He will make both denominators 15 and change the numerators of each fraction to match what he did on the bottom. The new fractions will be 2-5/15 + 12/15.)

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Explain how to rewrite the subtraction problem to make like terms for the problem 1-9/12 – 4/8 and then solve.

(Use the LCM of 24 to rewrite the denominators as like terms; 1-18/24 – 12/24.)

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Kelly solved the multiplication 5/8 x 1/5. She found the answer 5/40. What is the GCD of these terms and how can Kelly use it to put the answer in simplest form?

(Divide both the numerator and denominator by 5 to rewrite the fraction in simplest form as 1/8.)

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Mr. Garcia wants to purchase packs of juice and snack cakes for the 6^{th} grade social. The juices come in packs of 4 and the snack cakes come in packs of 8. He wants to buy equal amounts of juices and snack packs. What is the least number of packages of each that Mr. Garcia can buy.

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Water balloons come in packages of 50 and water guns come in packages of 4. Mrs. Jones Bill wants to have the same number of balloons and water guns for the end-of-year water fight. What is the least number of packages of balloons and water guns she needs to buy so that she has equal amounts of each?

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Enrique has to play soccer on August 7^{th} and every seventh day after that. How many days will he play soccer in August?

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Hotdogs are sold in packages of 8 and buns are sold in packages of 12. What is the least number of hot dogs and buns that Luis can buy so that he has the same number of each?

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Stephanie has to baby-sit on June 1^{st} and every 5^{th} day after that. How many days will she baby-sit in June?