how to find the least common denominator
To find the least common denominator (LCD), which is the least common multiple (LCM) of the denominators, you can either list out the multiples of each denominator and find the smallest common one, or use the prime factorization method by breaking down each denominator into its prime factors, identifying the unique prime factors with their highest powers, and then multiplying them together to find the LCD.
Method 1: Listing Multiples
This method is straightforward and works well for smaller numbers:

List the multiples of each denominator.
Example: For fractions with denominators 4 and 6, list their multiples:
Multiples of 4: 4, 8, 12, 16, 20, 24...
Multiples of 6: 6, 12, 18, 24...
Identify the smallest common multiple.
Example: In the lists for 4 and 6, the smallest number that appears in both lists is 12.
The smallest common multiple is the LCD.
Example: Therefore, the LCD of 4 and 6 is 12.

Method 2: Using Prime Factorization
This method is efficient for larger numbers:

Find the prime factors of each denominator. You can use a factor tree to help with this.
Example: For denominators 8 and 20:
8 = 2 × 2 × 2
20 = 2 × 2 × 5
Identify the unique prime factors that appear in any of the factorizations.
Example: The unique prime factors are 2 and 5.
For each unique prime factor, take the highest power that it appears in any of the factorizations.
Example: The factor 2 appears three times in 8, but only twice in 20. So, you will use 2 × 2 × 2 (three times). The factor 5 appears once in 20.
Multiply these factors together to find the LCD.
Example: 2 × 2 × 2 × 5 = 40. The LCD of 8 and 20 is 40.