Grade 6

1.04 Multiples and least common multiple (LCM)

Ideas

Least common multiple (LCM)

Least common multiple (LCM)

A multiple is the result of multiplying a number by an integer.

Remember, some numbers share multiples. For example, 12 is a common multiple of 2 and 3 because 2\times6=12 and 3\times4=12.

When we are asked to find the least common multiple, we are being asked to find the smallest multiple that is shared by two numbers.

Let's say we find the LCM of 6 and 8.

We can list the multiples of 6 and 8 as follows:

6:\, 6, 12, 18, 24, 30, 36
8:\, 8, 16, 24, 32, 40

Using a Venn diagram, we can see the multiples of 6 and 8 listed in each circle as two sets. The intersecting part represents the similarity or intersection between the two sets.

A Venn Diagram showing two overlapping circles labeled Multiples of 6 and Multiples of 8 inside a rectangle. The overlapping area has the number 24. The non-overlapping area of the circle labeled Multiples of 6 has the numbers 6, 12, 18, 30, and 36. The non-overlapping part of the circle labeled Multiples of 8 has numbers 8, 16, 32, and 40.

The Venn diagram shows that the least common multiple of 6 and 8 is 24.

Examples

Example 1

What is the least common multiple of 7 and 4?

Worked Solution

Create a strategy

We can consider the first seven multiples of 7 and the first seven multiples of 4, and choose the least number that appears in both lists.

Apply the idea

The multiples of 7 are: 7,\,14,\,21,\,28,\,35,\,42,\,49

The multiples of 4 are:4,\,8,\,12,\,16,\,20,\,24,\,28

The least number in both list is the LCM, 28.

Example 2

Use the following Venn diagram to find the LCM of 2,\,3, and 4

A Venn Diagram composed of 3 overlapping circles labeled Multiples of 2, Multiples of 3, and Multiples of 4. The part where the 3 circles overlap has 12 and 24 written on it. Aside from the overlap containing 12 and 24, the overlapping part between Multiples of 2 and Multiples of 3 has 6 and 18; between Multiples of 2 and Multiples of 4 has 4, 8, 16, and 20. The non-overlapping area of Multiples of 2 contains 2, 14, and 22. The non-overlapping area of Multiples of 3 contains 3, 9, 15, and 21.

Worked Solution

Create a strategy

When we are asked to find the least common multiple, we are being asked to find the smallest multiple that is shared by two or more numbers.

Apply the idea

The Venn diagram shows the three subsets of the mulitiples of 2, 3 and 4. The two multiples common to all three numbers are 12 and 24. The least common multiple, therefore is 12.

Idea summary

The least common multiple of two or more numbers is the smallest multiple that is shared by those numbers.

Outcomes

6.NS.B.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.