Quick answer
Factor each denominator into primes, take each prime at its highest power in any factorization, multiply to get lcm = LCD.
Formula
- 12 = 2² × 3
- 18 = 2 × 3²
- LCD = 2² × 3² = 36
Introduction
When multiples lists become long, prime factorization organizes the search.
The method scales to three or more denominators without rewriting huge charts.
Each prime factor tree should be neat enough that you can circle the highest powers quickly.
Review the general rule in the least common denominator formula before working through the trees below.
Prime factorization method
Break each denominator into prime factors. Example: 20 = 2²×5.
List every prime that appears in any factorization. For 12 and 20, primes are 2, 3, and 5.
Take the highest exponent of each prime across all factorizations. Here 2², 3¹, 5¹.
Multiply those powers to get lcm. LCD = 4×3×5 = 60 for denominators 12 and 20.
The lcm concept behind this process is developed further in least common multiple and LCD for students who want more context on multiples first.
Step-by-step breakdown
- Factor each denominator fully
- Record prime powers in a table
- Multiply highest powers
A table helps when three denominators appear together. Rows are primes; columns are denominators.
Division method alternative: divide pairs by primes until coprime, multiply divisors and leftovers. It matches prime thinking.
After LCD = 60 for 12 and 20, scale 5/12 = 25/60 and 7/20 = 21/60.
Procedure
- Factor denominators Use trees or repeated division on each bottom number.
- Select highest powers One row per prime that appears anywhere.
- Multiply for lcm Product equals LCD for the set.
- Scale numerators Build equivalent fractions at the LCD.
- Complete the operation Add, subtract, or compare as the problem requires.
Example: 7/12 and 5/18
12 = 2²×3 and 18 = 2×3². LCD = 2²×3² = 36.
Equivalents: 7/12 = 21/36 and 5/18 = 10/36.
Confirm at /#calculator. Try the same factor table on another denominator pair from your homework.