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6.NS.B.4Common CoreMathThe Number SystemGrade 6

6.NS.B.4: Greatest Common Factor, Least Common Multiple and the Distributive Property

In plain English: 6.NS.B.4 is the Common Core grade 6 math standard that asks students to find the greatest common factor of two whole numbers up to 100 and the least common multiple of two whole numbers up to 12. Students also use the distributive property to rewrite a sum such as 36 + 8 as 4 (9 + 2), a common factor times a sum of two numbers with no common factor.

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).

Common Core State Standards for Mathematics · Domain: The Number System (NS) · Cluster: Compute fluently with multi-digit numbers and find common factors and multiples.
Also written as 6.NS.4 · Official standard
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01

Lesson Plan

60-70 min

Overview

Students learn three connected skills. A factor of a whole number divides it with no remainder (3 is a factor of 12), and a multiple of a number is that number times a whole number (24 is a multiple of 12). The greatest common factor (GCF) of two numbers is the largest factor they share, and the least common multiple (LCM) is the smallest nonzero multiple they share. Students find the GCF of two whole numbers up to 100 and the LCM of two whole numbers up to 12, as the standard sets out, and decide which one a word problem needs.

Then students use the distributive property, which says that multiplying a sum by a number is the same as multiplying each part by it: 4 × (9 + 2) = 4 × 9 + 4 × 2. Read backward, it lets students pull a common factor out of a sum. The official example rewrites 36 + 8 as 4 (9 + 2). When the factor pulled out is the GCF, the two numbers left inside the parentheses have no common factor except 1, which is the form the standard asks for. Area models on grid paper make this visible. This work prepares for writing equivalent expressions in 6.EE.A.3.

Learning Objectives

By the end of this lesson, students will be able to:

  • List factors and find the greatest common factor of two whole numbers less than or equal to 100
  • List multiples and find the least common multiple of two whole numbers less than or equal to 12
  • Decide whether a word problem calls for the GCF or the LCM, and explain why
  • Use the distributive property to write a sum of two whole numbers as a common factor times a sum of two numbers with no common factor

Prior Knowledge Required

Students should already be comfortable with:

  • Finding all factor pairs of a whole number from 1 to 100, and telling prime from composite numbers 4.OA.B.4
  • Using the distributive property to multiply, such as 8 × 7 = 8 × 5 + 8 × 2 3.OA.B.5
  • Finding common denominators with equivalent fractions 5.NF.A.1
  • Multiplication facts through 12 × 12 and division with no remainder

Lesson Procedure

60-70 minutes of class time across 5 phases.

  1. Warm-Up10 minutes

    Pose a sharing problem and let students work with any method: drawings, lists or counters.

    Warm-Up Prompt

    "You have 18 cookies and 24 brownies for a class party. You want to make plates that are all exactly the same, with nothing left over. How many plates could you make? What is the greatest number of plates, and what goes on each one?"

    Collect answers: 1, 2, 3 or 6 plates all work, because each of these numbers divides both 18 and 24. The greatest is 6 plates, each with 3 cookies and 4 brownies. 4 plates do not work, because 18 cookies do not split evenly into 4 groups. Name the ideas: 1, 2, 3 and 6 are the common factors of 18 and 24 (factors both numbers share), and 6 is the greatest common factor. Point out that 3 and 4, the amounts on each plate, have no common factor except 1. That fact comes back in Part 3.

  2. Direct Instruction20-25 minutes

    Part 1: Greatest common factor. List the factors of each number in pairs, starting with 1 and the number itself, and stop when the pairs meet. Circle the factors that appear in both lists; the largest circled number is the GCF. For larger numbers, some students prefer prime factorization (writing a number as a product of primes, numbers whose only factors are 1 and themselves). The GCF is the product of the primes the two numbers share. Either method is fine; the standard does not require one.

    • GCF in a sharing problem

      A teacher has 42 pencils and 56 erasers. She wants to make identical supply kits with none left over. What is the greatest number of kits, and what is in each kit?

      Equation: Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42. Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56. GCF = 14, so 14 kits, each with 3 pencils and 4 erasers

    • GCF of two larger numbers

      Find the GCF of 84 and 90.

      Equation: 84 = 2 × 2 × 3 × 7 and 90 = 2 × 3 × 3 × 5; they share one 2 and one 3, so GCF = 2 × 3 = 6

    • LCM in a packaging problem

      Juice boxes come in packs of 6 and granola bars in packs of 8. What is the smallest number of each you can buy to have the same number of juice boxes and bars?

      Equation: Multiples of 8: 8, 16, 24; 24 is the first that is also a multiple of 6, so LCM = 24: 4 packs of juice and 3 packs of bars

    • LCM in a schedule problem

      Bus A leaves the station every 9 minutes and Bus B every 12 minutes. Both leave at 7:00. When do they next leave together?

      Equation: Multiples of 12: 12, 24, 36; 36 is the first multiple of 9 in the list, so LCM = 36 and they leave together at 7:36

    • Distributive property (official example)

      Express 36 + 8 as a multiple of a sum of two whole numbers with no common factor.

      Equation: GCF of 36 and 8 is 4: 36 + 8 = 4 × 9 + 4 × 2 = 4 (9 + 2); 9 and 2 have no common factor except 1

    Part 2: Least common multiple. List the multiples of the larger number and stop at the first one that the smaller number also divides. Diagram 2 shows the bus example on two number lines: the jumps of 9 and the jumps of 12 first land on the same number at 36. A common error is to multiply the two numbers: 9 × 12 = 108 is a common multiple, but not the least one. Remind students that this standard keeps the LCM to numbers 12 or less, so listing multiples is quick.

    Part 3: GCF or LCM? If a problem splits things into equal groups with nothing left over, the answer divides both numbers, so it is a common factor (often the GCF). If a problem waits for two repeating events to line up, or buys packs until the totals match, the answer is a common multiple (often the LCM).

    Part 4: The distributive property. Work the official example with Diagram 1: a rectangle 4 units tall and 9 + 2 units wide has an area of 36 + 8 = 44 square units, and also 4 × 11 = 44. So 36 + 8 = 4 (9 + 2). The number outside the parentheses is a common factor, and the numbers inside are what is left after dividing by it. Use the GCF so the numbers inside have no common factor. For example, 45 + 60 = 5 (9 + 12) is true, but 9 and 12 still share a factor of 3. Using the GCF 15 gives 45 + 60 = 15 (3 + 4), which is the form the standard asks for.

  3. Guided Practice15 minutes

    Pairs solve four problems. For each one, they first write whether it needs the GCF, the LCM or the distributive property, then solve and share.

    Guided practice problems and answers
    ProblemToolAnswer
    1. Find the GCF of 24 and 88.GCF8
    2. Find the LCM of 4 and 10.LCM20
    3. Write 30 + 42 as a multiple of a sum of two whole numbers with no common factor.Distributive property6 (5 + 7)
    4. A florist has 36 roses and 90 tulips. She makes identical bouquets with none left over. What is the greatest number of bouquets?GCF18 bouquets, each with 2 roses and 5 tulips

    In Problem 3, check the answer both ways: 6 × 5 + 6 × 7 = 30 + 42 = 72, and 5 and 7 have no common factor except 1. Listen for students who write 2 (15 + 21) or 3 (10 + 14). Both are true, but the numbers inside still share a factor.

  4. Independent Practice10-15 minutes

    Students solve five problems on their own. (1) The GCF of 32 and 80 (16). (2) The LCM of 6 and 9 (18). (3) The LCM of 5 and 12 (60, the product, because 5 and 12 have no common factor except 1). (4) Write 54 + 81 as a multiple of a sum of two whole numbers with no common factor (27 (2 + 3)). (5) One light blinks every 6 seconds and another every 10 seconds. They just blinked together. How many seconds until they blink together again? (30 seconds.)

  5. Closure5 minutes

    Exit ticket: (1) Find the GCF of 45 and 72 (9). (2) Find the LCM of 8 and 12 (24). (3) Write 20 + 35 as a multiple of a sum with no common factor (5 (4 + 7)). Then ask: "How can you tell, without computing, whether a problem needs the GCF or the LCM?"

Differentiation Strategies

For Struggling Students

  • Give a multiplication chart (1-12) so students can read multiples across a row and find factors by searching the chart
  • Have students build factor pairs as rectangles on grid paper: every rectangle with an area of 24 squares shows a factor pair of 24
  • For the distributive property, give a sentence frame: "___ + ___ = GCF × (___ + ___)", and have students check by multiplying back

For Advanced Students

  • Ask students to find two numbers under 50 with a GCF of 8 and an LCM of 48, and explain their search (an extension: this standard's LCM work stops at 12)
  • Ask why the GCF times the LCM of two numbers equals their product, testing it with three pairs of their own
  • Ask students to write 60 + 90 + 150 as a multiple of a sum of three numbers with no common factor (an extension beyond this standard, which uses two numbers)

Assessment Guidance

What to Look For

Check that students list factors in pairs so none are missed, and that they choose the greatest common factor, not just any common factor. For the LCM, watch for students who multiply the two numbers every time; ask them to list multiples to see if a smaller one works. In word problems, students should say why a problem needs the GCF (splitting into equal groups) or the LCM (events or packs lining up). In distributive property problems, check that students multiply back to confirm the sum and that the two numbers inside the parentheses have no common factor except 1.

02

Classroom Activities

3 Activities

1

Factor Rainbow Pairs

15 minPairs

Each pair gets 6 number-pair cards. For each number, students draw a factor rainbow: they write the factors in order and connect each factor pair with an arc. Then they circle the common factors and find the GCF.

Number-Pair Cards (6 cards)

  • 12 and 18 (GCF 6)
  • 30 and 50 (GCF 10)
  • 36 and 48 (GCF 12)
  • 35 and 64 (GCF 1)
  • 28 and 70 (GCF 14)
  • 45 and 75 (GCF 15)

Procedure

  • Partner A draws the rainbow for the first number and Partner B for the second
  • Together, circle every factor that appears in both rainbows in the same color
  • Write the GCF on a sticky note and check it by dividing both numbers by it

Discussion Questions

  • Which card has a GCF of 1? (35 and 64.) What does that tell you about their factors?
  • Which card has the largest GCF? (45 and 75, with 15)
  • How does the rainbow show that you have found every factor?

Modification for Distance Learning

Share the cards on a slide. Students type factor lists into a shared two-column table and highlight the common factors in the same color.

2

Hundred Chart Multiples Hunt

15 minPairs

Each pair gets a hundred chart and 8 cards with two numbers from 2 to 12. For each card, one partner shades the multiples of the first number and the other circles the multiples of the second. The first number that is both shaded and circled is the LCM.

LCM Cards (8 cards)

  • 3 and 5 (15), 4 and 6 (12), 6 and 10 (30), 8 and 12 (24)
  • 7 and 3 (21), 2 and 9 (18), 5 and 10 (10), 11 and 4 (44)

Procedure

  • Use a new hundred chart, or a new color, for each card
  • Record the LCM and the product of the two numbers side by side
  • Sort the cards into two piles: LCM equals the product, and LCM is less than the product

Discussion Questions

  • On which card is the LCM one of the two numbers? (5 and 10.) Why?
  • Which cards have an LCM equal to the product? (3 and 5, 7 and 3, 2 and 9, 11 and 4.) What do the two numbers on each of these cards have in common?
  • Why is the product always a common multiple, even when it is not the least one?

Challenge Variation

Pairs find the LCM of three numbers, such as 2, 3 and 4, on one chart, and explain how they know it is the least.

3

Area Models for Sums

20 minPairs

Pairs draw area models on grid paper to rewrite sums with the distributive property. The height of each rectangle is the GCF of the two numbers, and the two widths are the numbers left inside the parentheses.

Sum Cards (6 cards)

  • 24 + 40 = 8 (3 + 5)
  • 18 + 27 = 9 (2 + 3)
  • 56 + 21 = 7 (8 + 3)
  • 50 + 75 = 25 (2 + 3)
  • 16 + 60 = 4 (4 + 15)
  • 33 + 77 = 11 (3 + 7)

The teacher's copy shows the answers; students' cards show only the sums.

Procedure

  • Find the GCF of the two numbers and draw a rectangle with that height
  • Split the rectangle into two parts whose areas are the two numbers, and label each width
  • Write the sum as GCF × (width + width), then check that the two widths have no common factor except 1
  • Mistake check: a student wrote 48 + 36 = 4 (12 + 9). Is it true? Is it finished? (True, but 12 and 9 share a factor of 3, so it is not finished: 48 + 36 = 12 (4 + 3))

Discussion Questions

  • Which card has the tallest rectangle? (50 + 75, with a height of 25)
  • On the 16 + 60 card, why is 2 (8 + 30) true but not finished?
  • How does the area model show that both sides of the equation are equal?

03

Diagrams & Visual Aids

2 diagrams

Diagram 1: An Area Model for the Distributive Property

Area model: 36 + 8 written as 4 × (9 + 2) 9 2 4 4 × 9 = 36 4 × 2 = 8 36 + 8 = 4 × 9 + 4 × 2 = 4 × (9 + 2) = 4 × 11 = 44 The shared height 4 is the GCF of 36 and 8. The widths 9 and 2 have no common factor except 1, so this is the form the standard asks for.
The official example, drawn to scale on a grid. A rectangle 4 units tall is split into widths of 9 and 2, so its two parts have areas of 36 and 8. The whole rectangle is 4 units by 11 units, so 36 + 8 = 4 (9 + 2) = 44. The height 4 is the GCF of 36 and 8.

Diagram 2: Finding the LCM on Number Lines

Bus A leaves every 9 minutes and Bus B every 12 minutes, both at 7:00 Bus A 0 9 18 27 36 45 54 63 72 Bus B 0 12 24 36 48 60 72 36 minutes: the LCM Minutes after 7:00. The first time both buses leave together again is 7:36.
Two number lines count minutes after 7:00, drawn to the same scale. Bus A lands on the multiples of 9 and Bus B on the multiples of 12. The first number on both lines is 36, the LCM of 9 and 12, so the buses next leave together at 7:36. The next shared multiple is 72.

04

Homework Assignment

~30 min

6.NS.B.4 Homework: GCF, LCM and the Distributive Property

Directions: Show your factor lists or multiple lists for every GCF and LCM. For word problems, write whether you used the GCF or the LCM and why. For distributive property problems, check your answer by multiplying back.

Part 1: Greatest Common Factor (Problems 1-2)

  1. A school library has 72 fiction books and 90 nonfiction books to pack into boxes. Every box must have the same number of fiction books and the same number of nonfiction books, with none left over. (a) What is the greatest number of boxes? (b) How many of each kind of book go in each box?
  2. Find the GCF of each pair: (a) 26 and 39 (b) 100 and 75 (c) 17 and 51.

Part 2: Least Common Multiple (Problems 3-4)

  1. Maya waters her fern every 10 days and her cactus every 12 days. She watered both plants today. In how many days will she water both plants on the same day again?
  2. Find the LCM of each pair: (a) 3 and 8 (b) 7 and 12 (c) 6 and 12. For which pair is the LCM less than the product of the two numbers?

Part 3: The Distributive Property (Problems 5-6)

  1. Write each sum as a multiple of a sum of two whole numbers with no common factor: (a) 28 + 63 (b) 45 + 30 (c) 80 + 100.
  2. Deshawn wrote 60 + 48 = 4 (15 + 12). Is his equation true? Is it in the form the standard asks for? Explain, write the correct form and draw an area model for it.

Rubric

CriterionFull Credit (2 pts)Partial Credit (1 pt)No Credit (0 pts)
Greatest Common FactorComplete factor lists and correct GCFs, with the box contents foundA common factor that is not the greatest, or one list incompleteGCF incorrect or missing
Least Common MultipleCorrect LCMs from multiple lists, and the pair with LCM below the product namedCorrect but not least common multiples, such as the productLCM incorrect or missing
Distributive PropertyGCF outside, numbers with no common factor inside, checked by multiplyingTrue equation that is not finished, or no checkEquation not true or missing
ReasoningExplains why each word problem needs the GCF or the LCMNames the tool without a reasonNo explanation

05

Quiz: 20 Questions

Interactive, with answers

Instructions

Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.

Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.

0 of 20 answered · 0 correct

  1. Question 1 of 20 · Multiple Choice

    What is the greatest common factor of 16 and 40?

  2. Question 2 of 20 · Multiple Choice

    What is the greatest common factor of 63 and 42?

  3. Question 3 of 20 · Multiple Choice

    Ms. Ortiz has 52 red beads and 78 blue beads. She makes identical bags of beads with none left over. What is the greatest number of bags she can make?

  4. Question 4 of 20 · Multiple Choice

    Which pair of numbers has a greatest common factor of 1?

  5. Question 5 of 20 · Multiple Choice

    What is the greatest common factor of 96 and 60?

  6. Question 6 of 20 · Multiple Choice

    What is the least common multiple of 8 and 10?

  7. Question 7 of 20 · Multiple Choice

    A baker has two timers. One rings every 5 minutes and the other every 6 minutes. Both ring at 3:00. When will they next ring at the same time?

  8. Question 8 of 20 · Multiple Choice

    What is the least common multiple of 7 and 8?

  9. Question 9 of 20 · Multiple Choice

    What is the least common multiple of 3 and 12?

  10. Question 10 of 20 · Multiple Choice

    Which question is answered by finding a least common multiple?

  11. Question 11 of 20 · Multiple Choice

    Which expression shows 20 + 50 as a multiple of a sum of two whole numbers with no common factor?

  12. Question 12 of 20 · Multiple Choice

    Which is equal to 27 + 45 and uses two numbers with no common factor inside the parentheses?

  13. Question 13 of 20 · Multiple Choice

    Carlos says that 64 + 40 = 8 (8 + 5). Which statement is true?

  14. Question 14 of 20 · Multiple Choice

    A rectangle is 6 units tall. It is split into two parts, 7 units wide and 4 units wide. Which sum shows the area of the two parts?

  15. Question 15 of 20 · Short Answer

    Find the greatest common factor of 44 and 99. Show your factor lists.

  16. Question 16 of 20 · Short Answer

    Jada has soccer practice every 6 days and a piano lesson every 7 days. Both are today. In how many days will they fall on the same day again?

  17. Question 17 of 20 · Short Answer

    Write 56 + 72 as a multiple of a sum of two whole numbers with no common factor. Check by multiplying.

  18. Question 18 of 20 · Short Answer

    Write 15 + 100 as a multiple of a sum of two whole numbers with no common factor.

  19. Question 19 of 20 · Short Answer

    An art teacher has 32 markers and 56 sheets of paper. She makes identical art kits with none left over. What is the greatest number of kits, and what is in each kit?

  20. Question 20 of 20 · Short Answer

    For each question, say whether you need the GCF or the LCM, then answer it. (a) Muffins come in boxes of 4 and bananas in bunches of 5. What is the least number of each you can buy to have exactly one banana per muffin? (b) A board 35 cm by 49 cm is covered with equal square tiles, with no gaps and no cutting. What is the side length of the largest tile that works?

0 of 20 answered · 0 correct

06

Frequently Asked Questions

10 Questions

What does 6.NS.B.4 mean?

6.NS.B.4 means students can find the greatest common factor of two whole numbers up to 100, find the least common multiple of two whole numbers up to 12, and use the distributive property to pull a common factor out of a sum. The official example rewrites 36 + 8 as 4 (9 + 2).

Why does 6.NS.B.4 limit the LCM to numbers 12 or less?

The standard sets these limits: GCF problems use whole numbers up to 100, and LCM problems use whole numbers up to 12. With small numbers, students can find the LCM by listing multiples or using multiplication facts they already know, so the focus stays on understanding what a least common multiple is.

What is the difference between the GCF and the LCM?

The GCF is the largest number that divides both numbers, so it is never larger than the smaller number. The LCM is the smallest nonzero number that both numbers divide, so it is never smaller than the larger number. For 10 and 15, the GCF is 5 and the LCM is 30.

How do you know whether a word problem needs the GCF or the LCM?

Ask what the answer must do. If it must divide both amounts into equal groups with nothing left over, such as identical gift bags, use a common factor, usually the GCF. If it must be reached by counting up in steps of both numbers, such as two schedules lining up or packs of different sizes matching, use a common multiple, usually the LCM.

How is the distributive property used in 6.NS.B.4?

Students use it backward to pull a common factor out of a sum. In the official example, 36 = 4 × 9 and 8 = 4 × 2, so 36 + 8 = 4 × 9 + 4 × 2 = 4 (9 + 2). The standard asks for two numbers with no common factor inside the parentheses, so the factor pulled out should be the GCF.

Do students need prime factorization for 6.NS.B.4?

No. The standard does not name a method. Listing factors and multiples works well for these numbers. Prime factorization is a useful option for larger numbers: 70 = 2 × 5 × 7 and 98 = 2 × 7 × 7 share 2 and 7, so their GCF is 14.

What are common mistakes with GCF and LCM?

Students sometimes mix up the two, or give a common factor that is not the greatest. Another frequent error is to multiply two numbers to get the LCM: 6 × 8 = 48 is a common multiple, but the LCM is 24. With the distributive property, a common mistake is to stop at a factor that is not the GCF, such as 2 (18 + 4) for 36 + 8, where 18 and 4 still share 2.

What is the GCF of two numbers that share no factors?

It is 1, because 1 is a factor of every whole number. For example, 9 and 25 have a GCF of 1. Such numbers are called relatively prime. Their LCM is their product, so the LCM of 4 and 9 is 36.

How does 6.NS.B.4 prepare students for algebra?

Pulling a common factor out of a sum is the first step toward factoring expressions. In grade 6, students write equivalent expressions such as 3 (2 + x) = 6 + 3x (6.EE.A.3). In grade 7, they factor expressions like 4x + 12 = 4 (x + 3) (7.EE.A.1), and in high school they rewrite expressions using their structure (HSA.SSE.A.2).

How can parents practice GCF and LCM at home?

Use everyday sharing and scheduling. Ask how many identical snack bags you can make from a box of crackers and a bag of grapes. Ask when two chores on different schedules, such as every 3 days and every 4 days, will fall on the same day. Let your child explain whether they used a common factor or a common multiple.