The Least Common Denominator
 The Least Common Multiple
A multiple of a number is a whole number
times that number. For example, some multiples of 6 are
6, 12, 18,
24, 30, and 36
If two numbers are given then a common
multiple of the two numbers is a number
that is a multiple of both. Of all the common multiples of two
numbers, there is a smallest one which we call the least
common multiple.
Example
Find the least common multiple of 6
and 9.
Solution
One way of solving this problem is to write out multiples of each and see
what is common to the list:
6,
12, 18, 24, 30, 36, ... multiples
of 6
9, 18, 27, 36, 45,
... multiples
of 9
We see that the numbers 18
and 36 are both
common multiples of 6
and 9. The
least common multiple is the smallest which is 18.
Example
Find the least common multiple of 8
and 32.
Solution
Instead of listing many multiples of each, we just notice that 32
is a multiple of 8
and hence 32 is a
common multiple. It is the first multiple of 32.
We can conclude that 32 is the
least common multiple of 8
and 32.
In general, the least common multiple of two
numbers with one the multiple of the other is just the larger number.
Exercise
 Find the least common multiple of 15
and 54.
Hold mouse over the yellow rectangle for the solution
 Find the least common multiple of 9
and 81.
Hold mouse over the yellow rectangle for the solution
As you saw from the Exercise A, writing out many multiples of each
number can be tedious. There is an alternate method that may save
time. The strategy is based on the following idea. A
multiple of a number is a multiple of each of the prime
divisors.
Steps in Finding the LCM
 Write the prime factorization of each number
 List the primes that occur in at least one of
the factorizations
 Form a product using each prime the greatest
number of time it occurs in any one of the expressions
Example
Find the LCM of 45
and 21
Solution
 45 = 9 x 5 = 3 x 3 x 5
21 = 3 x 7
 3, 5, and 7
 3 x 3 x 5 x 7
The prime 3 occurs two times as it does in
3 x 3 x 5
= 9 x 5 x 7 = 45 x
7 = 315
Exercises
Find the LCM of
 18 and 40
Hold mouse over the yellow rectangle for
the solution
 12 and 15
Hold mouse over the yellow rectangle for
the solution
 27 and 10
Hold mouse over the yellow rectangle for
the solution
 The Least Common Denominator
We define the least common denominator
of two fractions as the least common multiples of the denominators.
Examples
Find the least common denominator of
 3/4 and 9/10
 5/6 and 10/11
Solutions
 We find the least common multiples of 4
and 10
4 = 2 x
2
10 = 2 x 5
So the least common denominator is
2 x 2 x 5 = 20
 We find the least common multiples of 6
and 11
6
= 2 x 3 11
is prime
So the least common denominator is
2 x 3 x 11 = 66
Exercises
Find the least common denominator of
 3/14 and 2/63
Hold mouse over the yellow rectangle for the solution
 8/25 and 23/100
Hold mouse over the yellow rectangle for the solution
 Building Up Fractions With a Least Common
Denominator
We have already learned how to simplify a fraction by dividing through by a
common factor. Sometimes it is convenient to be able to work this
process in reverse.
Example
Build up the fraction to answer the question
5 ?
=
6 24
Solution
We see that
24 = 6 x 4
so
5 5 x
4 20
=
=
6 6 x
4 24
Exercise
Build up the fraction to answer the question
3 ?
=
7 35
Hold mouse over the yellow rectangle for the
solution
Example
Which number is larger: 5/8
or 9/14?
Solution
Since the denominators are different, these numbers are difficult to
compare. Our strategy is to build up each fraction to fractions with
the least common denominator. We first find the least common
denominator:
8 = 2 x 2 x 2 14
= 2 x 7
The least common denominator is
2 x
2 x 2 x 7 = 56
The next step is to notice that
8 x
7 = 56
and 14
x 4 = 56
We write
5 5 x
7 35
=
=
8 8 x
7 56
and
9 9
x 4 36
=
=
14 14 x
4 56
Since
35 36
<
56 56
We conclude that
5 9
<
8 14
Exercise
Which is larger:
3/10 or 7/25?
Hold mouse over the yellow rectangle for the
solution
Example
Write the three fractions
1/6, 5/8 and 3/10 as
equivalent fractions with the LCD as the denominators.
Solution
We have
6 = 2 x 3
8 = 2 x 2 x 2 10
= 2 x 5
So the least common denominator is
2 x 2 x 2 x 3 x 5 = 8
x 3 x 5 = 24 x 5 = 120
We write
1 1 x
20 20
=
=
6 6 x
20 120
5 5 x
15 75
=
=
8 8 x
15 120
3 3
x
12
36
=
=
10 10 x
12 120
Exercise
Write the three fractions
2/15, 4/9 and 3/25
as equivalent fractions with the LCD as the
denominators.
Hold mouse over the yellow rectangle for the
solution
Back to the Fractions page
Back to the Math 187A
page
email
Questions and Suggestions
