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GCF and LCM
Lesson 3.01
After completing this lesson, you will be able to say:
• I can find the least common multiple of two
whole numbers.
• I can find the greatest common factor of two
whole numbers.
• I can use the distributive property to rewrite the
sum of two numbers using the greatest common
factor.
Key Words
Prime number:
A whole number greater than one that has exactly two factors, the
number 1 and itself.
The numbers 2, 3, 5, 7, 11, 13, and 17 are all examples of prime
numbers.
Composite number:
A number with more than two factors.
The number 12 is a prime number because we can break it into more
than 2 factors: 2 x 6, 3 x 4, 1 x 12
Prime factorization
Prime factorization:
Writing any composite number as a product of its prime factors.
• Every number can be written as a product of its factors.
• You can also write any composite number as a product of its prime factors,
which is called prime factorization.
• When you write the prime factorization, it is always a good idea to write the
factors in order from least to greatest.
Prime factorization
If you’ve done factorization before, you may have used factor trees to
find the prime factorization of a number
Because division is the opposite of multiplication, you
can start the factorization process by asking, “What two
numbers multiply to equal 36?”
36
Now ask yourself, "Can 12 or 3 break down into smaller
numbers, or are they prime?”
3 is a prime number so let’s circle it
What about the number 12? Can we break it down into
smaller parts?
12
3
2
6
Okay, 12 can be broken down into 6 times 2. Are either
of those numbers prime?
2 is prime so let’s circle it
How can we break down 6?
Finally, 6 can be factored into 3 × 2. Both numbers will
go into our prime factorization of 36.
Prime factorization of 36: 2 x 2 x 3 x 3
Try it!
Create a factor tree, and write the prime factorization of 64
Check you work
64
4
2
16
2
4
4
Prime factorization of 64: 2 x 2 x 2 x 2 x 2 x 2
Least Common Multiple
Multiple:
A number that is created when it is multiplied by
other numbers.
Least common multiple:
The smallest of the common multiples between
two or more numbers, also known as the LCM.
Finding the Least Common Multiple
Finding the LCM from a list
Find the LCM of 2 and 3
Step 1: Identify the multiples of the two numbers
2, 4, 6, 8, 10, 12, 14, 16, 18
3, 6, 9, 12, 15, 18, 21, 24
Step 2: Identify the common multiples
2, 4, 6, 8, 10, 12, 14, 16, 18
3, 6, 9, 12, 15, 18, 21, 24
Step 3: Identify the least common multiple (LCM)
6 is the smallest and is the LCM.
Finding the Least Common Multiple
Find the LCM using prime factorization
Find the LCM of 4 and 6
Find the prime factorization of 4 and 6
Identify the factors in prime factorization that they have common.
4=2×2
6=2×3
Both 4 and 6 have a factor of 2 in common.
Calculate the LCM
Both numbers have one 2 in common, so it will be used only one time.
There is also an extra factor of 2 and an extra factor of 3 that do not appear in both prime
factorizations. Multiply the common factor with all extra factors.
LCM = 2 × 2 × 3 = 12
The LCM of 4 and 6 is 12.
Try It
What is the LCM of 8 and 10?
Check your work
What is the LCM of 8 and 10?
Greatest Common Factor
Greatest common factor:
The largest factor shared in common
by two or more numbers, also known
as the GCF.
Finding the GCF
Finding the GCF using a list
Find the GCF of 12 and 18
Step 1: Identify the factors of the two numbers
12 : 1, 2, 3, 4, 6, 12
18: 1, 2, 3, 6, 9, 18
Step 2: Identify the common factors
12: 1, 2, 3, 4, 6, 12
18: 1, 2, 3, 6, 9, 18
Step 3: Identify the Greatest Common Factor (GCF)
6 is the largest and is the GCF
Finding the GCF
Finding the GCF using prime factorization
The GCF of 36 and 54
Find the prime factorization of 36 and 54
2x3x3x3
Identify the factors in prime factorization that they have common.
36 = 2 x 2 x 3 x 3
54 = 2 x 3 x 3 x 3
The greatest common factor is the product of all of the common prime
factors.
GCF = 2 × 3 × 3 = 18
Try It
Find the GCF of 48 and 96
Check your work
48 =
2x2x2x2x3
96 = 2 x 2 x 2 x 2 x 2 x 3
GCF = 2 x 2 x 2 x 2 x 3
GCF = 48
Using the GCF with the Distributive Property
One use of the GCF is with the distributive property. First, recall the
distributive property with this visual below to see how 4 x (2 + 6) is the same
as 8 + 24.
The distributive property shows the 4 rows of 2 yellow boxes plus 6 green
boxes is the same as 4 rows of 2 yellow boxes plus 4 rows of 6 green boxes.
What is happening is that the 4 is being distributed by multiplying it with each
number being added in the parentheses to get 4 × 2 + 4 × 6
Using the GCF with the Distributive Property
Use the greatest common factor and the distributive property to express 44 + 16 in
a different way
First, identify the GCF of both numbers by listing the factors
The factors of 44: 1, 2, 4, 11, 22, 44
The factors of 16: 1, 2, 4, 8, 16
The GCF is 4.
Then rewrite the problem using the distributive property
Since the GCF is 4, you can divide both 44 and 16 by 4. When you do this, place the
GCF on the outside and the two quotients remain in parentheses.
44 + 16 = 4 × (11 + 4)
Also, 4 × (11 + 4) can be written as 4(11 + 4).
So how do you know this is correct?
Just calculate the sum.
44+16
=
4(11+4)
44+16
=
4(15)
60
=
60
Both sides are equal.
Try It
How can you rewrite the sum of 18 + 48 using the GCF?
Check your work
How can you rewrite the sum of 18 + 48 using the GCF?
First, identify the GCF of both numbers by listing the factors
The factors of 18: 1, 2, 3, 6, 9, 18
The factors of 48: 1, 2, 3, 4, 6, 8, 12,16, 24, 48
The GCF is 6.
Then rewrite the problem using the distributive property
Since the GCF is 6, you can divide both 18 and 48 by 6. When you do this, place the
GCF on the outside and the two quotients remain in parentheses.
18 + 48 = 6 × (3 + 8)
Also, 6 × (3 + 8) can be written as 6 (3 + 8)
Now that you completed this lesson, you should
be able to say:
• I can find the least common multiple of two
whole numbers.
• I can find the greatest common factor of two
whole numbers.
• I can use the distributive property to rewrite
the sum of two numbers using the greatest
common factor.