Greatest Common Factor and Least Common Denominator
Greatest Common Factor – [Also called the Greatest Common Divisor.] The GCF of a
group of expressions is a product of the largest number of factors that can be found in all
of the expressions simultaneously.
Note: The GCF must be able to divide into each of the expressions in the group evenly.
Hence, the power of a prime factor in the GCF cannot be larger than the smallest power
of the same prime factor appearing in the group of expressions.
Common use: The GCF is commonly used when factoring terms.
Finding the GCF: First, factor all of the expressions completely. Write each expression
as a product of prime factors (prime numbers and prime polynomials) raised to powers.
Second, list all of the basic factors found in every expression. Third, for each basic
factor, find the least power occurring in the expressions (if there is an expression that
does not contain a certain prime factor, then the effective power of that prime factor is
zero.) Finally, the GCF is the product of the prime factors raised to the powers found in
the last step.
Least Common Denominator – The LCD is the Least Common Multiple of a group of
denominators. The Least Common Multiple of a group of expressions is a product
containing the smallest number of factors such that every expression in the group will
divide evenly into it.
Note: The LCD (or LCM) must be such that every expression in the group can divide
evenly into the LCD. Hence, the power of a prime factor in the LCD must be at least as
large as the largest power of the same prime factor appearing in the group of expressions.
Common use: The LCD is commonly used when one is trying to create a common
denominator in order to add or subtract rational expressions. Also, the LCD is used when
dealing with a rational equation. Multiplying both sides of the equation by the LCD and
simplifying each side will create an equation with no fractions.
Finding the LCD: First, factor all of the expressions completely. Write each expression
as a product of prime factors (prime numbers and prime polynomials) raised to powers.
Second, list all of the basic factors found in every expression. Third, for each basic
factor, find the largest power occurring in the expressions. Finally, the LCD is the
product of the prime factors raised to the powers found in the last step.
Example – Given the numbers 48, 90, 120, and 180.
Factor each as a power of primes.
48 = 2 4 ⋅ 3
90 = 2 ⋅ 32 ⋅ 5
120 = 23 ⋅ 3 ⋅ 5
180 = 2 2 ⋅ 32 ⋅ 5
List the basic factors: 2, 3, 5
For the GCF, find the least power of each basic factor appearing in the expressions.
Notice that 5 does not appear as a factor in 48 so the power of 5 should be zero.
That is, 48 = 2 4 ⋅ 3 ⋅ 50 .
Basic Factor
2
Power in
48
4
Power in
90
1
Power in
120
3
Power in
180
2
Least power
appearing
1
3
1
2
1
2
1
5
0
1
1
1
0
Finally, the GCF is the product of the basic factors raised to the least powers found
above. Thus, the GCF is 21 ⋅ 31 ⋅ 50 or simply 6. [Notice, 6 divides into each of the
numbers evenly.]
For the Least Common Multiple (such as the LCD if the expressions are denominators),
find the largest power of each basic factor appearing in the expressions.
Basic Factor
2
Power in
48
4
Power in
90
1
Power in
120
3
Power in
180
2
Largest power
appearing
4
3
1
2
1
2
2
5
0
1
1
1
1
Finally, the Least Common Multiple (such as the LCD if the expressions are
denominators) is the product of the basic factors raised to the largest powers found above.
Thus, the Least Common Multiple is 2 4 ⋅ 32 ⋅ 51 or simply 720. [Notice, each of the
numbers divides into 720 evenly.]
Example – Given the expressions 12 x 5 ( x + 1) 2 , 24 x3 ( x 2 − 1) , and 48 x 4 ( x − 1) .
Factor each as powers of primes and prime polynomials.
12 x 5 ( x + 1) 2 = 223 x5 ( x + 1) 2
24 x 3 ( x 2 − 1) = 233x3 ( x + 1)( x − 1)
48 x 4 ( x − 1) = 243 x 4 ( x − 1)
List the basic factors: 2, 3, x, (x + 1), (x – 1)
For the GCF, find the least power of each basic factor appearing in the expressions.
Basic Factor
2
Power in
12 x 5 ( x + 1) 2
2
Power in
24 x3 ( x 2 − 1)
3
Power in
48 x 4 ( x − 1)
4
Least power
appearing
2
3
1
1
1
1
x
5
3
4
3
(x +1)
2
1
0
0
(x – 1)
0
1
1
0
Finally, the GCF is the product of the basic factors raised to the least powers found
above. Thus, the GCF is 2231 x3 ( x + 1)0 ( x − 1)0 = 12 x3 .
For the Least Common Multiple (such as the LCD if the expressions are denominators),
find the largest power of each basic factor appearing in the expressions.
Basic Factor
2
Power in
12 x 5 ( x + 1) 2
2
Power in
24 x3 ( x 2 − 1)
3
Power in
48 x 4 ( x − 1)
4
Largest power
appearing
4
3
1
1
1
1
x
5
3
4
5
(x +1)
2
1
0
2
(x – 1)
0
1
1
1
Finally, the Least Common Multiple (such as the LCD if the expressions are
denominators) is the product of the basic factors raised to the largest powers found above.
Thus, the Least Common Multiple is 2431 x 5 ( x + 1) 2 ( x − 1)1 = 48 x5 ( x + 1) 2 ( x − 1) .