# Lesson 17

Common Multiples

### Lesson Narrative

In this lesson, students use contextual situations to learn about common multiples and the least common multiples of two whole numbers. They develop strategies for finding common multiples and least common multiples.

### Learning Goals

Teacher Facing

• Comprehend (orally and in writing) the terms “multiple,” “common multiple,” and “least common multiple.”
• Explain (orally and in writing) how to calculate the least common multiple of 2 whole numbers.
• List the multiples of a number and identify common multiples for two numbers in a real-world situation.

### Student Facing

Let’s use multiples to solve problems.

### Required Preparation

For the first classroom activity, "The Florist's Order," provide access to two different colors of snap cubes (100 of each color) to students who would benefit from manipulatives. For students with visual impairment, provide access to manipulatives that are distinguished by their shape rather than by color.

### Student Facing

• I can explain what a common multiple is.
• I can explain what the least common multiple is.
• I can find the least common multiple of two whole numbers.

### Glossary Entries

• common factor

A common factor of two numbers is a number that divides evenly into both numbers. For example, 5 is a common factor of 15 and 20, because $$15 \div 5 = 3$$ and $$20 \div 5 = 4$$. Both of the quotients, 3 and 4, are whole numbers.

• The factors of 15 are 1, 3, 5, and 15.
• The factors of 20 are 1, 2, 4, 5, 10, and 20.
• common multiple

A common multiple of two numbers is a product you can get by multiplying each of the two numbers by some whole number. For example, 30 is a common multiple of 3 and 5, because $$3 \cdot 10 = 30$$ and $$5 \cdot 6 = 30$$. Both of the factors, 10 and 6, are whole numbers.

• The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33 . . .
• The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40 . . .

The common multiples of 3 and 5 are 15, 30, 45, 60 . . .

• greatest common factor

The greatest common factor of two numbers is the largest number that divides evenly into both numbers. Sometimes we call this the GCF. For example, 15 is the greatest common factor of 45 and 60.

• The factors of 45 are 1, 3, 5, 9, 15, and 45.
• The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
• least common multiple

The least common multiple of two numbers is the smallest product you can get by multiplying each of the two numbers by some whole number. Sometimes we call this the LCM. For example, 30 is the least common multiple of 6 and 10.

• The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 . . .
• The multiples of 10 are 10, 20, 30, 40, 50, 60, 70, 80 . . .

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