Lesson 35

Slide 1Lesson 35

Testing for DivisibilityWO.17 Use long division to determine if one number

is divisible by another.

WO.23 Use divisibility rules to determine if a number is divisible by 2, 3, 5, or 9 and understand the justification for these rules.

Chapter 7

Lesson 35

Slide 2Lesson 35

Objectives

• Understand and use the divisibility rules for 2, 3, 5 and 9.

Slide 3Lesson 35

Remember from Before

• What is a factor?

• What is a multiple?

• How are multiples and factors related?

Slide 4Lesson 35

Get Your Brain in Gear

1. Use mental math to divide 369 by 9.

2. Use mental math to divide 85 by 5.41

17

Slide 5Lesson 35

Multiples of 2.

All multiples of 2 can be expressed as the repeated addition of 2.

10 = 2 + 2 + 2 + 2 + 2

Slide 6Lesson 35

Is 36 divisible by 2? Let’s try to express 36 as repeated addition of 2.

Slide 7Lesson 35

Let’s try 21.

We have a unit square left over.

This means that 21 is not divisible by 2.

Slide 8Lesson 35

What about larger powers of 10?

100 = 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10

100 = 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2

Since all the powers of ten are multiples of 10, they also are all multiples of 2.

Slide 9Lesson 35

Divisible by 2 rule:

If a whole number ends in 0, 2, 4, 6 or 8, then the number is divisible by 2. Otherwise it is not divisible by 2.

Slide 10Lesson 35

Applying the rule, is the following number divisible by 2?

47,297,593

The digit in the 100 place is 3, and 3 is not divisible by

2.

Slide 11Lesson 35

Check for Understanding

1. Determine whether the number is divisible by 2.

a. 23

b. 78

c. 504

d. 8,241

e. 6,794

Not divisible by 2.

Divisible by 2.

Not divisible by 2.

Divisible by 2.

Divisible by 2.

Slide 12Lesson 35

Divisibility by 5 Is 10 divisible by 5?

10 = 5 + 5

Since 10 is divisible by 5, so are all the larger powers of 10.

Slide 13Lesson 35

Divisibility by 5 rule:If a whole number ends in 0 or 5, then the number is divisible by 5. Otherwise it is not divisible by 5.

According to this rule, would 365 be divisible by 5?

Slide 14Lesson 35


2. Determine if the number is divisible by 5.

a. 70

b. 553

d. 72865

c. 10003

e. 8003000 Divisible by 5.

Divisible by 5.

Divisible by 5.

Not divisible by 5.

Not divisible by 5.

Slide 15Lesson 35

Divisibility by 9Since 10 is not divisible by 9, we can’t simply check the last digit.

Let’s see if 27 is divisible by 9:

27 = 9 + 9 + 9

Slide 16Lesson 35

When testing for divisibility by 9, we see that each 10 leaves 1 left over, so we can treat each 10 as a 1.

Is 52 divisible by 9?

Since 5 + 2 equals 7, we conclude 52 is not divisible by 9.

Slide 17Lesson 35


Remember, each 10 is treated as a 1.

Since 6 + 3 equals 9, this means 63 is divisible by 9.


How do you know?

Slide 18Lesson 35

7 + 5 + 6 = 18Since 18 is divisible by 9, we conclude that 756 is also divisible by 9.

What about larger numbers? Is 756 divisible by 9?

Slide 19Lesson 35

If the digits of a whole number add up to a multiple of 9, then the number is divisible by 9. Otherwise it is not divisible by 9.

Slide 20Lesson 35


3. Determine whether the number is divisible by 9.

a. 73

b. 108

c. 7812

d. 6873

e. 98016

Not divisible by 9.

Not divisible by 9.

Not divisible by 9.

Divisible by 9.

Divisible by 9.

Slide 21Lesson 35

Let’s develop a test for divisibility by 3.

Let’s check if 42 is divisible by 3.

Slide 22Lesson 35

If the digits of a whole number add up to a multiple of 3, then the number is divisible by 3. Otherwise it is not divisible by 3.


5 + 9 + 2 = 16

1 + 6 = 7

Slide 23Lesson 35

We can verify that 592 is not divisible by 3 using long division:

Slide 24Lesson 35

Check for Understanding4. Test whether the number is divisible by 3. Verify the result using long division.

5. Using what you learned in this lesson, how can you quickly determine if 1,335 is divisible by 15? Is it?

6. What is the smallest number you can add to 7,120 to make it divisible by 3?

7. When you divide 2,349,684 by 5, will there be a remainder? What will the remainder be?

a. 84 b. 275 c. 1086

d. 23938 e. 62505

Yes Yes

YesNo

No

You check to see if it is divisible by 3 and divisible by 5. Thus 1,335 is divisible by 15.

Add 2. 7,122 is divisible by 3.

Yes; 4

Slide 25Lesson 35

Multiple Choice Practice

1. Which of the following numbers is NOT a factor of 29,910?

2

3

5

9

Slide 26Lesson 35

A student made the following claims about divisibility. What is the student misunderstanding? What would you tell this student to correct their understanding?

Find the Errors

The student was able to correctly determine if a number is divisible by 2 or 5, but misunderstood how to test for divisibility of 3. You cannot in general look at the last digit to determine if it is divisible by 3, you must add all the digits together and check if the number is a multiple of 3. 5 + 2 + 3 = 10, which is not a multiple of 3. Therefore, 523 is not divisible by 3.

Technology

Lesson 35