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B2.6 · Find greatest common factors and least common multiples

Learn to find greatest common factors and least common multiples through clear examples and targeted practice.

Ontario Grade 7 Mathematics

Number

Grade 7 Mathematics — B2.6

Factors and multiples help us describe how whole numbers fit together. For example, factors can help divide a group into equal-sized groups with no leftovers. Multiples can help find when repeating counts meet at the same number. In this lesson, you will find the greatest common factor and least common multiple of whole numbers. You will use lists and check that your answers fit the numbers in the problem.

What you will learn

  • Identify factors and multiples of whole numbers.
  • Find the greatest common factor of two whole numbers.
  • Find the least common multiple of two whole numbers.
  • Choose whether a problem calls for a greatest common factor or a least common multiple.

1. Review: factors and multiples

A whole number is a number such as 0, 1, 2, 3, and so on. For this lesson, we will list factors and multiples of positive whole numbers.
A factor of a number divides it into equal groups with nothing left over. For example, 3 is a factor of 15 because 15 can be shared into 3 equal groups of 5. The number being divided is also a factor: 15 groups of 1 make 15.
A multiple is the result of multiplying a number by a whole number. You can make a list of positive multiples by counting in equal steps. Multiples of 4 begin at 4 and go up by 4 each time.
These two ideas describe different things. Factors of a number are never greater than that number. Multiples can keep getting larger.
3×5=153\times5=15
  • A factor divides a number evenly, with no remainder.
  • A multiple is found by multiplying a number by a whole number.

2. Find the greatest common factor

A common factor of two or more numbers is a factor they share. The greatest common factor, or GCF, is the largest factor they share.
To find a GCF, list the factors of each number. Circle or note the factors that appear in every list. Then choose the largest shared factor. The list is a useful model because it shows that no shared factor has been missed.
For example, the factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. Their common factors are 1, 2, 3, and 6. The greatest of these is 6.
A GCF is useful when a quantity needs to be divided into equal groups or equal parts without leftovers. The factor must fit both numbers. Choosing the greatest shared factor gives the largest group size that works for both.
GCF⁡(18,30)=6\operatorname{GCF}(18,30)=6
  • List all the factors of each number.
  • Find the factors that appear in every list.
  • Select the largest common factor.

3. Find the least common multiple

A common multiple of two or more numbers is a number that appears in each number's multiples. The least common multiple, or LCM, is the smallest positive common multiple.
To find an LCM, list positive multiples of each number in order. Look for the first number that appears in every list. It is the least common multiple because the lists are in increasing order.
For example, the multiples of 6 are 6, 12, 18, 24, and 30. The multiples of 8 are 8, 16, 24, and 32. The first shared multiple is 24.
An LCM is useful when equal repeating counts need to reach the same total. The shared number must be a multiple of each starting number. The least one is the first shared total, rather than a later one.
LCM⁡(6,8)=24\operatorname{LCM}(6,8)=24
  • List positive multiples in order.
  • Find the first multiple that appears in every list.
  • Check that the answer is a multiple of each starting number.

4. Choose and check

Read what the problem asks for. If it asks for the largest equal group size that fits into given amounts, look for a GCF. If it asks when repeated counts first reach the same total, look for an LCM.
The words largest and least matter. A common factor is not automatically the greatest common factor. A common multiple is not automatically the least common multiple. Compare the shared choices before deciding.
You can check a GCF by confirming that it divides each number evenly and that no larger factor is shared. You can check an LCM by confirming that it is a multiple of each number and that no smaller positive common multiple comes first.
Try these: Find the GCF of 20 and 28. Find the LCM of 5 and 7. Decide which to find if two equal-size groups must be made from 16 red counters and 24 blue counters, with no counters left over. The examples below show how to organize this work.
  • Largest equal group size points to a GCF.
  • First shared total points to an LCM.
  • Check both the shared property and the word greatest or least.

Worked example

Find a GCF from factor lists

Find the greatest common factor of 24 and 36.
  1. List the factors
    Write every whole-number factor of each number. A factor must divide its number evenly, so include factor pairs that multiply to the number.
    24:1,2,3,4,6,8,12,24;36:1,2,3,4,6,9,12,18,3624: 1,2,3,4,6,8,12,24;\quad 36: 1,2,3,4,6,9,12,18,36
  2. Find the shared factors
    Compare the lists and keep the numbers that appear in both. These are the common factors.
    1,2,3,4,6,121,2,3,4,6,12
  3. Choose the greatest
    The greatest number in the shared list is 12. It divides both original numbers evenly, so it is their greatest common factor.
    GCF⁡(24,36)=12\operatorname{GCF}(24,36)=12
Answer: The greatest common factor of 24 and 36 is 12.
Check: Both 24 and 36 divide evenly by 12. The shared factor list shows that none of their common factors is larger than 12.

Worked example

Find an LCM from multiples

Find the least common multiple of 9 and 12.
  1. List multiples
    Count by 9 and by 12. Continue until you find a number that appears in both lists.
    9,18,27,36,45;12,24,36,489,18,27,36,45;\quad 12,24,36,48
  2. Find the first shared number
    The number 36 appears in both lists. The earlier entries do not match, so 36 is the first positive common multiple.
    3636
  3. Check the result
    The answer is a multiple of both numbers: it is 4 groups of 9 and 3 groups of 12. Since the lists are in order and show no earlier match, it is the least common multiple.
    LCM⁡(9,12)=36\operatorname{LCM}(9,12)=36
Answer: The least common multiple of 9 and 12 is 36.
Check: 36 is divisible by 9 and by 12. The ordered lists show that it is their first positive shared multiple.

Worked example

Choose the right idea

A club has 18 green counters and 30 yellow counters. It wants to make the greatest possible number of identical sets, using every counter. How many sets can it make?
  1. Decide what to find
    The question asks for the greatest number of identical sets. Each set must divide both counter totals evenly. This calls for a greatest common factor.
    GCF⁡(18,30)\operatorname{GCF}(18,30)
  2. List common factors
    The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. Their shared factors are 1, 2, 3, and 6.
    1,2,3,61,2,3,6
  3. Use the greatest shared factor
    The greatest shared factor is 6, so the club can make 6 identical sets. Each set has 3 green counters and 5 yellow counters. These amounts use all the counters and are equal in every set.
    18÷6=3,30÷6=518\div6=3,\quad 30\div6=5
Answer: The club can make 6 identical sets.
Check: Six sets use all 18 green counters and all 30 yellow counters. The factor lists show that no larger number of identical sets can divide both totals evenly.

Common mistakes and how to avoid them

Choosing a factor that divides only one of the numbers.
Correction: A common factor must divide every number in the question evenly.
Stopping at any shared factor or multiple.
Correction: For a GCF, choose the greatest shared factor. For an LCM, choose the least positive shared multiple.
Mixing up factors and multiples.
Correction: Factors divide into a number. Multiples are found by counting in equal steps from a number.

Lesson summary

  • A factor divides a number evenly; a multiple is made by multiplying a number by a whole number.
  • The GCF is the greatest factor shared by the numbers.
  • The LCM is the least positive multiple shared by the numbers.
  • Use lists to find the shared choices, then check greatest or least.

Check your understanding

Question 1

What is the greatest common factor of 16 and 24?
  1. 4
  2. 8
  3. 16
  4. 48
Show answer and explanation
8
The common factors are 1, 2, 4, and 8. The greatest is 8.

Question 2

What is the least common multiple of 4 and 10?
  1. 10
  2. 14
  3. 20
  4. 40
Show answer and explanation
20
Multiples of 4 include 4, 8, 12, 16, and 20. Multiples of 10 include 10 and 20. Their first positive shared multiple is 20.

Question 3

You need the greatest number of equal groups that can be made from 21 pencils and 28 pens, with none left over. Which should you find?
  1. The GCF of 21 and 28
  2. The LCM of 21 and 28
  3. A multiple of 21 only
  4. A factor of 28 only
Show answer and explanation
The GCF of 21 and 28
The number of equal groups must divide both amounts. The greatest number of groups is their greatest common factor.

Key terms

Factor
A whole number that divides another number evenly.
Multiple
A number found by multiplying a given number by a whole number.
Common factor
A factor shared by two or more numbers.
Greatest common factor (GCF)
The largest factor shared by two or more numbers.
Common multiple
A multiple shared by two or more numbers.
Least common multiple (LCM)
The smallest positive multiple shared by two or more numbers.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation B2.6. It is a study resource, not an official curriculum publication.

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