Divisibility with Venn Diagrams

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Divisibility with Venn Diagrams
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Notes

• We will be discussing sets of number that are
  divisible by a particular number. For instance,
  all numbers divisible by twelve. This set
  includes 12, 24, 36, 48, 60

• Note that the set of numbers divisible by twelve
  is also the set of multiples of twelve.

• When making Venn diagrams, always be precise in
  your labeling. Do not merely put a 12 as a label
  if you are talking about the numbers divisible by
  12. One reason is that you might easily get
  confused and start thinking of the factors (1, 2,
  3, 4, 6, 12) instead of the multiples of 12.

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How are the numbers divisible by 6 related to the
numbers divisible by 3?
Numbers Divisible by 3
Numbers Divisible by 6
The numbers divisible by 6 are a subset of the
numbers divisible by 3?
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Place the numbers 12, 30, 15, 6, 3, 2, 27 and 35
in the proper place in the Venn diagram. Click
when you are done.
2
12
15
6
3
30
35
27
5
How are the numbers divisible by 4 related to the
numbers divisible by 6?
Numbers Divisible by 6
Numbers Divisible by 4
Because there are some numbers, such as 24, which
are divisible by both 4 and 6, an intersection of
the two sets must be shown..
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Place the numbers 12, 30, 8, 6, 4, 3, 24, 35 and
36 in the proper place in the Venn diagram.
Click when you are done.
All Whole Numbers
3
6
4
12
24
30
8
35
36
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Can you think of two numbers, where the set of
numbers divisible by the first and the set of
numbers divisible by the second are disjoint as
shown below?.
Numbers Divisible by ?
Numbers Divisible by ?
THERE ARE NO SUCH NUMBERS!
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You may at first think that if you chose two
numbers with no common factors, that this would
be the case. For instance 5 and 6.
All Whole Numbers
30
However, when you multiply 5 times 6, you get a
common factor 30.
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For any two whole numbers, the relationship
between the set of numbers divisible by the first
and the set of numbers divisible by the second
will always fit one of the models below..
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Consider the validity of the following if, then
statement.
If a number is divisible by 24, then it is
divisible by 8.
Satisfy yourself that this is a true statement
and then draw a Venn diagram to accompany it.
The numbers divisible by 24 are a subset of the
numbers divisible by 8.
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Determine which statements are true and which are
false.

• If a number is divisible by 9, then it is
  divisible by 3.
• If a number is divisible by 6, then it is
  divisible by 12.
• If a number is divisible by 4, then it is
  divisible by 16.
• If a number is divisible by 16, then it is
  divisible by 4.

TRUE
FALSE
FALSE
TRUE
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Which of the following statements is true?
If a number is divisible by both 2 and 6, then it
is divisible by 12.
If a number is divisible by both 3 and 4, then it
is divisible by 12.
8
30
30
8
24
24
36
12
12
18
36
6
6
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To see this place theses numbers in both Venn
diagrams. 12, 24, 8,
6, 18, 36, 30
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The numbers 30, 18 and 6, in the intersection of
the two sets below provide counterexamples that
disprove the statement
Counterexample 18 is divisible by both 2 and 6
however, 18 is not divisible by 12.
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The Venn diagram and the true statement below
provide a divisibility rule for 12.
If a number is divisible by both 3 and 4, then it
is divisible by 12.
A number is divisible by 12 if it is divisible by
both 3 and 4.
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Which statement is false and which provides a
correct divisibility rule for 50? Trying to place
some numbers may help.
If a number is divisible by both 5 and 10, then
it is divisible by 50.
If a number is divisible by both 2 and 25, then
it is divisible by 50.
75
75
100
100
40
50
50
20
40
25
25
20
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Find a counterexample to show that the statement
below is false.
If a number is divisible by both 5 and 10, then
it is divisible by 50.
Counterexample 40 is divisible by both 5 and 10
however, 40 is not divisible by 50.
We also have a divisibility rule.
A number is divisible by 50 if it is divisible by
both 2 and 25.
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How can one tell which factors to use to make a
divisibility rule for a number?
Study the table below.
Notice that 2 and 6 have a common factor of 2. 5
and 10 have a common factor of 10. To make a
divisibility rule, one must use numbers that have
NO common factors.
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Make a divisibility rule for 24.
Prime factoring 24 we get 2 x 2 x 2 x 3.
To make a rule we must put all the 2s together.
This gives us 2 x 2 x 2 8 and 3.
If a number is divisible by 8 and 3, then it is
divisible by 24.
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Try this with 36.
Prime factoring 36 we get 2 x 2 x 3 x 3.
To make a rule we must put all the 2s together.
and all the 3s together
This gives us 2 x 2 4 and 3 x 3 9.
If a number is divisible by 4 and 9, then it is
divisible by 36.
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Try one more, 90.
Prime factoring 90 we get 2 x 3 x 3 x 5.
To make a rule we must put all the 3s together.
It doesnt matter where the 2 and 5 go!
This gives us 3 x 3 9 and 2 x 5 10.
Or 3 x 3 x 2 18 and 5.
Slow down
Or 3 x 3 x 5 45 and 2.
Slow down
If a number is divisible by 9 and 10, then it is
divisible by 90.
If a number is divisible by 18 and 5, then it is
divisible by 90.
If a number is divisible by 45 and 2, then it is
divisible by 90.
As long as you keep the 3s together, you are
good.
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Thats All
Right click and select End Show. Then Close to
return.