LCM: Least Common Multiple and LCD: Least Common Denominator

Contributed by:

Diego Tue, Jan 25, 2022 02:01 PM UTC

This pdf provides the basics of LCM (Least Common Multiple) and LCD (Least Common Denominator). The LCD and the LCM require the same math process: Finding a common multiple of two (or more) numbers. The only difference between LCD and LCM is that the LCD is the LCM in the denominator of a fraction. So, one could say that the least common denominators are a special case of least common multiples.

1. STRATEGIES FOR FINDING THE
LEAST COMMON MULTIPLE (LCM)/LEAST COMMON DENOMINATOR (LCD)
The least common multiple (LCM) of a given set of numbers is the smallest positive number divisible
by the numbers in the set. For example, if we list the multiples of 4 and 6, we can see these numbers
share common multiples of 12, 24, 36, and 48 to name a few.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, …
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, …
Even though 24, 36 and 48 are multiples of 4 and 6, the LCM is 12 because 12 is the smallest number
divisible by 4 and 6.
When we need to find a common denominator for a given set of fractions, the LCM is called the least
common denominator (LCD). To find the LCD of a given set of fractions, check the denominators of
the fractions:
STRATEGIES
1) Do the smaller denominators divide the larger? If they do, the larger denominator is the LCD.
3 1 5
EXAMPLE 1: Find the LCD of , , and
4 2 8
Because 8 is divisible by 4 and 2, the LCD = 8.
2) Are the denominators prime or relatively prime numbers? (Prime numbers are numbers divisible
only by themselves and 1; relatively prime numbers share no common factor.) When the
denominators are prime or relatively prime, multiply the denominators to find the LCD.
2 4 1
EXAMPLE 2: Find the LCD of , , and
3 5 2
The denominators of the fractions are prime numbers. To find the LCD, multiply the denominators:
LCD = 2 • 3 • 5 = 30.
3 5
EXAMPLE 3: Find the LCD of and
4 7
The denominators of the fractions are relatively prime numbers because they share no common
factors: 4 = 2 • 2 and 7 = 1 • 7. To find the LCD, multiply the denominators:
LCD = 4 • 7 = 28.
PBCC 1 SLC Lake Worth Math Lab

2. 2 4
EXAMPLE 4: Find the LCD of and
x 9
Because the value of "x" is unknown, the only factors of x are "1" and "x." This means that 9 and
"x" share no common factors, so the LCD = 9 • x.
3) If the largest denominator is not divisible by the smaller denominators, list the multiples of the
largest to find the LCD.
5 1 7 8
EXAMPLE 5: Find the LCD of , , , and
4 6 10 15
The smaller denominators do not divide the larger. As shown below, we find the LCD sooner when
we list the multiples of the largest denominator.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, …
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 64, 60, …
Multiples of 10: 10, 20, 30, 40, 50, 60, …
Multiples of 15: 15, 30, 45, 60, …
LCD = 60
4) If the LCD is not among the first 5 or 6 multiples you list, try prime factorization and a factor box.
5 2 7
EXAMPLE 6: Find the LCD of , , and
12 15 18
Step 1: Write the prime factorization of each denominator and list the factors in a table of primes,
as shown:
2 3 5
12 = 2 • 2 • 3 = 22 • 3 → 22 3
15 = 3 • 5 → 3 5
18 = 2 • 3 • 3 = 2 • 32 → 2 32
Step 2: Take the highest power of any factor the numbers share in common and any factor the
numbers do not share in common. The LCD is the product of these factors:
LCD = 22 • 32 • 5 = 180
PBCC 2 SLC Lake Worth Math Lab

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