MAT 030/060 – Lesson 6
The least common multiple and greatest common factor
Part 1: Finding the least common multiple (LCM)
The multiples of a number are all the products of that number and 1, 2,
3, 4, 5…
3x1=3
3x2=6
3x3=9
3 x 4 = 12
3 x 5 = 15
3 x 6 = 18
3 x 7 = 21
therefore, the multiples of 3 are: 3, 6, 9, 12, 15,
18, 21 ….
Keep in mind that multiples of a number are always equal to or larger
than that number.
A number that is a multiple of two or more numbers is a
common multiple of the numbers.
The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36…
The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42…
Some common multiples of 4 and 6 are: 12, 24 and 36.
The least common multiple(LCM) of two or more numbers is the
smallest common multiple of those numbers.
The LCM of 4 and 6 is 12.
Finding the LCM of small numbers is simple enough. Simply list the
multiples of the numbers and identify the smallest one.
When you need to find the LCM of large numbers, there is a simple
procedure to follow:
1. write the numbers with an inverted division
sign and find the smallest number that will
divide both evenly. Divide and write the solution
below.
2. Repeat this process until the numbers can
no longer be divided by the same number.
3. Multiply the outside numbers together to
get the LCM.
Practice:
Find the LCM of 8 and 14
Find the LCM of 24, 36 and 50
Find the LCM of 50, 84 and 135
Part 2: Finding the greatest common factor (GCF)
As you already know, factors of a number are the numbers that divide
that number equally.
The factors of 6 are: 1, 2, 3 and 6
Keep in mind that factors of a number are equal to or smaller than that
number.
A number that is a factor of two or more numbers is a common factor of
those numbers.
The factors of 30 are: 1, 2, 3, 5, 6, 10, 15 and 30.
The factors of 105 are: 1, 3, 5, 7, 15, 21, 35 and 105.
The common factors of 30 and 105 are: 1, 3, 5 and 15.
The greatest common factor (GCF) is the largest common factor of two
or more numbers.
The GCF of 30 and 105 is 15.
To find the GCF of numbers list all the factors of each number and
identify the largest common factor.
Example:
Find the GCF of 36 and 54
The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36
The factors of 54 are: 1, 2, 3, 6, 9, 18, 27, and 54
The numbers in bold are all common factors and 18 is the GCF
A simpler way to find the GCF is to prime factor both numbers
The prime factorization of 36 is 2 x 2 x 3 x 3
The prime factorization of 54 is 2 x 3 x 3 x 3
Notice that the prime factorization of 36 and 54 both have one 2 and
two 3s in common. So, we simply multiply these common prime factors
to find the GCF. Like this… 2 x 3 x 3 = 18
Practice
Find the GCF of 32, 56 and 72
Find the GCF of 126 and 180
Find the GCF of 11, 24 and 30