FIVE METHODS for FINDING the LCM or LCD
Method 1
FINDING THE LCD by INSPECTION
Sometimes you can determine the LCD by simply examining the original denominators.
Determine the LCD by inspection:
1/4 and 5/6
12 ← LCM
1/3 and 2/5
15 ← LCM
Method 2
FINDING THE LCD by the THREE-RULE Method
1. Locate the largest denominator in the problem. Check to see if it is the LCM. If the
smaller denominator(s) divide evenly into it, it is the LOWEST COMMON
DENOMINATOR.
2. Multiply the largest denominator by 2, 3, 4, 5, 6, 7, 8, and 9. Check each product to
see if the LCM has been found.
3. Multiply DIFFERENT denominators together. This product should be the LOWEST
COMMON DENOMINATOR.
Caution: You must apply these rules in order from first to third.
Find the LCM for 12 and 15.
NOTE: Applying rule 1, we see that 15 is the largest denominator, but 12 does not
divide evenly into it.
So, we apply rule 2:
2 x 15 = 30
but 30 is NOT divisible by 12.
3 x 15 = 45
but 45 is NOT divisible by 12.
4 x 15 = 60
and 60 IS divisible by 12. It is the Least
Common Denominator (LCM)
Method 3
FINDING THE LCD or (LCM) by using ONE LISTOF MULTIPLES
One method for finding LCM’s uses one list of multiples. Let’s consider finding the LCM
of 9 and 12. The largest number, 12, is not a multiple of 9; so we check multiples of 12
until we find a number that is also a multiple of 9:
1 • 12 = 12, not a multiple of 9;
2 • 12 = 24, not a multiple of 9;
3 • 12 = 36, a multiple of 9: 4 • 9 = 36
The LCM of 9 and 12 is 36.
To find the LCM of a set of numbers using a list of multiples:
a) Determine whether the largest number is a multiple of the others. If it is, it is the
LCM. That is, if the largest number has the others as factors, the LCM is that
number.
b) If not, check multiples of the largest number until you get one that is a multiple
of each of the others.
Find the LCM of 12, 18, and 40.
a) Identify 40 as the largest denominator. Neither 12 nor 18 will divide evenly
into 12, so 12 is NOT the LCM.
b)
2
3
4
5
x
x
x
x
40
40
40
40
=
=
=
=
80
120
160
200
9 x 40 = 360
NOT the LCM
NOT the LCM
NOT the LCM
NOT the LCM
6 x 40 = 240
7 x 40 = 280
8 x 40 = 320
NOT the LCM
NOT the LCM
NOT the LCM
and 360 is divisible by 12 and 18. It IS the LCM.
Method 4
FINDING THE LCD or (LCM) by using PRIME FACTORIZATION
Another method for finding LCM’s uses prime factorizations. Consider again 20 and
30. Their prime factorizations are 20 = 2 • 2 • 5 and 30 = 2 • 3 • 5. Let’s look at these
prime factorizations in order to find the LCM. Any multiple of 20 will have to have two
2’s as factors and one 5 as a factor. Any multiple of 30 will have to have one 2, one 3,
and one 5 as factors. The smallest number that will satisfy these conditions is
To find the LCM of a set of numbers using prime factorizations:
a) Find the prime factorization of each number.
b) Create a product of factors, using each factor the greatest number of times that
it occurs in any one factorization.
NOTE:
33 is part of the LCD because 33 was the highest degree 3 was used (in 27).
22 is part of the LCD because 22 was the highest degree 2 was used (in 84).
51 is part of the LCD because 51 was the highest degree 5 was used (in 90).
Method 5
FINDING THE LCD or (LCM) by using DIVISION BY PRIMES
Here is another method for finding LCM’s that is especially useful for three or more
numbers. For example, to find the LCM of 48, 72, and 80, we first look for the lowest
prime number that will divide evenly into at least two of the original numbers. Then we
divide as follows:
We repeat the process, (always using the lowest prime number that will divide into
at least two of the original numbers), bringing down any numbers not divisible by the
prime, until we can divide no more; that is until there are no two numbers divisible by
the same prime:
No two of 1, 3, and 5 are divisible by the same prime number. We stop here.
The LCM is
2 • 2 • 2 • 2 • 3 • 3 • 5 = 720
To find the LCM of a set of numbers using division by primes:
a) First look for the lowest prime number that will divide into at least two of the numbers
with no remainder. Then divide, bringing down any numbers not divisible by the
prime. NOTE: Keep using the lowest prime number until it no longer works. Then
test the next lowest prime number.
b) Repeat the process until you can divide no more, that is, until there are no two
numbers divisible by the same prime.