# Lesson 18

Using Common Multiples and Common Factors

### Lesson Narrative

In this lesson, students apply what they have learned about factors and multiples to solve a variety of problems. In the first activity, students to use what they have learned about common factors and common multiples to solve less structured problems in context (MP1). The two activities that follow are optional. The optional activity "More Factors and Multiples" allows students to explore common factors and common multiples of 3 whole numbers and present their findings. The optional activity "Factors and Multiples Bingo" allows students to practice finding multiples and factors.

### Learning Goals

Teacher Facing

• Choose to calculate the greatest common factor or least common multiple to solve a problem about a real-world situation, and justify (orally) the choice.
• Present (orally, in writing, and using other representations) the solution method for a problem involving greatest common factor or least common multiple.

### Student Facing

Let’s use common factors and common multiple to solve problems.

### Required Preparation

If using the optional activity, "More Factors and Multiples," provide access to tools for creating a visual display.

If using the optional "Factors and Multiples Bingo" activity, prepare one set of pre-printed slips, one set of pre-printed Bingo boards cut from the blackline master, and enough Bingo chips for the boards. Each group of 2 students will share 1 Bingo board. It is suggested that the Bingo boards are copied onto cardstock for durability.

### Student Facing

• I can solve problems using common factors and multiples.

### 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 . . .