Finding the Least Common Multiple or Denominator
Definition: The Least Common Multiple or Denominator (LCM or LCD) is the smallest
number/expression that two or more numbers or expressions into evenly.
For Example: For the numbers 6 and 9, 54 is a common multiple, but it is not the least
common multiple, since 18 is the smallest number 6 and 9 divides into
evenly.
Finding the LCD: One approach to finding the LCD is to look at the prime factors that
occurs the most (look for the largest exponent) in each of the numbers of expressions that
are involved. For instance, in the previous example, 6 and 9, factor each number into its
prime factors.
6 = 3●2
= 3●2
9 = 3●3
= 32
The largest exponent on the 3 is two and on the 2 is one; so the LCD is 32●2 = 18.
Another example of finding the LCD of 90 and 120:
90 = 2●3●3●5
= 2●32●5
120 = 2●2●2●3●5
= 23●3●5
Choose the largest exponent on each factor to get the LCD, which is 23●32●5 = 360.
Examples for Finding the LCD of Algebraic Expressions:
The same approach is used to find the LCD for algebraic expressions—factor into prime
factors first.
Example:
Find the LCD of 12x2y3w and 20x y2.
12x2y3w = 22●3●x2●y3●w
20x y2 = 22●5●x●y2
Choose the largest exponent of each factor, so the LCD is 22●3●5●x2●y3●w = 60x2y3w
Example:
Find the LCD of 3x3 + 6x2 and 6x2 – 24
3x3 + 6x2 = 3x2(x + 2)
= 3●x2●(x + 2)
6x2 – 24 = 6(x2 – 4)
= 2●3●(x + 2)●(x - 2)
The LCD is 2●3●x2●(x + 2)●(x - 2) = 6x2(x + 2)(x - 2)