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Chapter 7
Lesson 35
Testing for Divisibility
Lesson 35
WO.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.
Slide 1
Objectives
• Understand and use the divisibility rules for 2, 3,
5 and 9.
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Slide 2
Remember from Before
• What is a factor?
• What is a multiple?
• How are multiples and factors related?
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Slide 3
Get Your Brain in Gear
1. Use mental math to divide 369 by 9.
41
2. Use mental math to divide 85 by 5.
17
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Slide 4
Multiples of 2.
All multiples of 2 can be expressed as the
repeated addition of 2.
10 = 2 + 2 + 2 + 2 + 2
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Slide 5
Is 36 divisible by 2? Let’s try to express 36 as repeated
addition of 2.
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Slide 6
Let’s try 21.
We have a unit square left over.
This means that 21 is not divisible by 2.
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Slide 7
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.
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Slide 8
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.
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Slide 9
Applying the rule, is the following number divisible by
2?
47,297,593
0
The digit in the 10 place is 3, and 3 is not divisible by
2.
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Slide 10
Check for Understanding
1. Determine whether the number is divisible by 2.
a. 23
Not divisible by 2.
b. 78
Divisible by 2.
c. 504
Divisible by 2.
d. 8,241 Not divisible by 2.
e. 6,794 Divisible by 2.
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Slide 11
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.
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Slide 12
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?
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Slide 13
Check for Understanding
2. Determine if the number is divisible by 5.
a. 70
Divisible by 5.
b. 553
Not divisible by 5.
c. 10003
Not divisible by 5.
d. 72865
Divisible by 5.
e. 8003000 Divisible by 5.
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Slide 14
Divisibility by 9
Since 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
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Slide 15
Is 52 divisible by 9?
When testing for divisibility by 9, we see that each 10 leaves 1
left over, so we can treat each 10 as a 1.
Since 5 + 2 equals 7, we conclude 52 is not divisible by 9.
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Slide 16
Is 63 divisible by 9?
Remember, each 10 is treated as a 1.
Since 6 + 3 equals 9, this means 63 is divisible
by 9.
Is 85 divisible by 9?
How do you know?
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Slide 17
What about larger numbers? Is 756 divisible by 9?
7 + 5 + 6 = 18
Since 18 is divisible by 9, we conclude that 756 is also
divisible by 9.
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Slide 18
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.
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Slide 19
Check for Understanding
3. Determine whether the number is divisible by 9.
a. 73
Not divisible by 9.
b. 108
Divisible by 9.
c. 7812
Divisible by 9.
d. 6873
Not divisible by 9.
e. 98016 Not divisible by 9.
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Slide 20
Let’s develop a test for divisibility by 3.
Let’s check if 42 is divisible by 3.
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Slide 21
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.
Is 592 divisible by 3?
5 + 9 + 2 = 16
1+6= 7
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Slide 22
We can verify that 592 is not divisible by 3 using long
division:
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Slide 23
Check for Understanding
4. Test whether the number is divisible by 3. Verify the
result using long division.
a. 84
b. 275 No
Yes
d. 23938
No
c. 1086 Yes
e. 62505 Yes
5. Using what you learned in this lesson, how can you
quickly determine if 1,335 is divisible by 15? Is it?
You check to see if it is divisible by 3 and divisible by 5. Thus 1,335 is divisible by 15.
6. What is the smallest number you can add to 7,120 to
make it divisible by 3?
Add 2. 7,122 is divisible by 3.
7. When you divide 2,349,684 by 5, will there be a
remainder? What will the remainder be? Yes; 4
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Slide 24
Multiple Choice Practice
1. Which of the following numbers is NOT a factor of
29,910?
2
3
5
9
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Slide 25
Find the Errors
A student made the following claims about
divisibility. What is the student misunderstanding?
What would you tell this student to correct their
understanding?
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.
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Slide 26