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1.3 Common Factors and
Common Multiples
GOAL
Use prime factorization to identify common factors and common multiples.
Learn about the Math
Jordan and Reilly are creating a large square mural.
The mural will be made of 36 cm by 48 cm
rectangles covered with coloured squares. They
want these squares to be as large as possible,
measured in whole numbers of centimetres.
48 cm
36 cm
Then Jordan and Reilly plan to arrange copies of the 36 cm by 48 cm
rectangle to form a large square mural that measures the least possible
whole number of centimetres.
They decide to use the greatest common factor (GCF) and
least common multiple (LCM) of 36 and 48 to determine the
dimensions of both sizes of squares.
are the dimensions of the small squares and the
? What
large square mural?
greatest common
factor (GCF)
the greatest whole
number that divides
into two or more other
whole numbers with
no remainder; for
example, 4 is the
greatest common
factor of 8 and 12
least common
multiple (LCM)
the least whole
number that has two or
more given numbers
as factors; for example,
12 is the least common
multiple of 4 and 6
Example 1: Using a Venn diagram to identify the GCF
What are the dimensions of the small squares? Use the greatest common factor (GCF)
of 36 and 48.
Jordan’s Solution
36 2 2 3 3
48 2 2 2 2 3
Prime
factors
of 36
3
Then I arranged the prime factors in a Venn
diagram. The common prime factors are in the
overlap.
Prime
2
factors
2 of 48
2
3
2
First I wrote the prime factorization of 36 and 48.
I multiplied the common prime factors to
determine the other common factors of 36 and
48. So, 2, 3, 4 (2 2), 6 (2 3), and 12 (2 2 3)
are the common factors of 36 and 48.
The GCF is 2 2 3 12.
A 12 cm by 12 cm square is the largest possible square that divides a 36 cm by 48 cm rectangle.
12
Chapter 1
NEL
Example 2: Using a Venn diagram to identify the LCM
What are the dimensions of the final square mural? Use the least common multiple (LCM)
of 36 and 48.
Reilly’s Solution
36 2 2 3 3
48 2 2 2 2 3
Prime
factors
of 36
3
2
Prime
factors
2 of 48
Then I arranged the prime factors in a Venn
diagram.
I multiplied all the prime numbers in both circles
to determine the LCM of 36 and 48.
2
3
First I wrote the prime factorization of 36 and 48.
2
144 cm
48 cm
The LCM is 3 2 2 3 2 2 144.
The final mural will be a 144 cm by
144 cm square.
36 cm
144 cm
Reflecting
1. How did identifying the GCF and LCM help Jordan and Reilly
decide on the dimensions of the small squares and the large square
mural?
2. Why do you think Jordan multiplied the factors in the overlap of the
Venn diagram to determine other common factors of 36 and 48?
3. Why do you think Reilly multiplied the numbers in the three
sections of the Venn diagram to determine the LCM of 36 and 48?
NEL
Number Relationships
13
Work with the Math
Example 3: Using Venn diagrams to identify the GCF and LCM
Show how to use prime factorization to identify the GCF and LCM of each pair of numbers.
a) 27 and 42
b) 18 and 35
Solution A
Solution B
Write the prime factorization of 27 and 42.
27 3 3 3
42 2 3 7
Write the prime factorization of 18 and 35.
18 2 3 3
35 5 7
Record the prime factors in a Venn diagram.
Record the prime factors in a Venn diagram.
Prime
factors
of 27
3
3
3
Prime
factors
of 42
2
7
The GCF is the product of the numbers in
the overlap. The GCF of 27 and 42 is 3.
The LCM is the product of the numbers in
both circles. The LCM of 27 and 36 is
3 3 3 2 7 378.
A
Checking
4. Identify the GCF and LCM of each pair of
numbers.
a) 120 2 2 2 3 5
210 2 3 5 7
b) 252 2 2 3 3 7
60 2 2 3 5
5. a) Identify another common factor of
each pair of numbers in question 4.
b) Identify another common multiple of
each pair of numbers in question 4.
14
Chapter 1
Prime
factors
of 18
3 2
3
Prime
factors
of 35
5
7
The GCF is the product of the numbers in
the overlap. There are no prime factors in the
overlap, but 1 is a common factor of both
18 and 35. So, the GCF of 18 and 35 is 1.
The LCM is the product of the numbers in
both circles. The LCM of 18 and 35 is
2 3 3 5 7 630.
B
Practising
6. Use prime factorization to identify at least
three common factors and at least three
common multiples of each pair of numbers.
a) 48 and 60
b) 32 and 64
c) 24 and 32
d) 512 and 648
7. Identify the GCF and LCM of each pair of
numbers.
a) 78 2 3 13
442 2 13 17
b) 32 2 2 2 2 2
24 2 2 2 3
NEL
8. Use prime factorization to identify four
common factors and four common
multiples of each pair of numbers.
a) 468 13
396 22 32 11
b) 840 23 3 5 7
2000 24 53
c) 1818 2 32 101
606 2 3 101
22
13. Given the GCF or LCM, what else do you
know about each pair of numbers?
a)
b)
c)
d)
32
14. Explain how you can use these centimetre
bars to identify the common factors and
GCF of 6 and 8.
9. Identify the GCF and LCM of each pair of
numbers.
a) 64 and 240
b) 55 and 275
6
Prime
Prime
5
factors
factors
of 360 2 2 of 480
2
2
3
3
2
15. Explain how to identify the GCF and LCM
of a pair of numbers, if one number is a
factor of the other number.
16. What pairs of numbers fit this description?
List as many pairs as you can.
“A pair of numbers has a sum of 100.
One number is a multiple of 3. The other
number is a multiple of 11.”
b) Identify four other common factors.
c) Identify four other common multiples.
11. A rectangle measures 72 cm by 108 cm.
a) A 2 cm by 2 cm square can be used to
cover the rectangle without any spaces
or overlapping. Explain why.
b) How can you use the common factors
of 72 and 108 to identify all the squares
that can be used to cover the rectangle?
c) List all other squares with wholenumber dimensions that can be used to
cover the rectangle.
12. Identify the GCF and LCM of each pair of
numbers.
NEL
8
c) 48 and 72
d) 120 and 200
10. a) Identify the GCF and LCM of 360
and 480.
a) 40 and 48
b) 120 and 400
Two numbers have a GCF of 2.
Two numbers have an LCM of 2.
Two numbers have a GCF of 3.
Two numbers have an LCM of 10.
c) 101 and 200
d) 1024 and 1536
C
Extending
17. The prime factorizations of two numbers, a
and b, have some missing prime factors.
a23■
b■■
a) The GCF of a and b is 5. What is the
value of a?
b) The LCM of a and b is 210. What is
the value of b?
18. Show how you can use prime factorization
and a Venn diagram to identify the GCF
and LCM of 12, 48, and 64.
Number Relationships
15