How to Find the Least Common Denominator (LCD): Step-by-Step Method
Finding the LCD of two or more fractions takes three steps: list multiples, find the smallest match, rewrite each fraction. This guide shows the method with worked examples, plus a prime-factorization shortcut for larger numbers.
The least common denominator (LCD) is the smallest number that every denominator in a set of fractions divides into evenly. It is a special case of the least common multiple, applied to the bottoms of fractions. You need it to add, subtract, or compare fractions.
Skip the arithmetic with our LCD calculator. Paste your fractions and get the LCD plus the rewritten fractions back. The method below is what the calculator does under the hood.
What the LCD actually is
The least common denominator is the smallest number that every denominator in your set divides into evenly. It is a specific case of the least common multiple, applied to the bottoms of fractions. A few quick ones show the pattern.
You need the LCD when adding, subtracting, or comparing fractions. You do not need it when multiplying or dividing them, so see find the product for those rules.
Method 1: listing multiples (best for small numbers)
This works for any set of denominators but gets slow above the low twenties. List the multiples of each denominator, pick the smallest value that appears in every list, then rewrite each fraction over that number. The shared multiple is your LCD.
Then rewrite each fraction so its bottom is the LCD. Multiply the top and bottom of each fraction by whatever turns its denominator into the shared value, and the fractions are ready to add.
Method 2: prime factorization (fast for bigger numbers)
Break each denominator into prime factors, take the highest power of every prime that shows up, and multiply those together. This avoids long multiple lists and scales to large denominators where listing would take forever.
Then take the highest power of each prime and multiply. The highest power of two and the highest power of three across both factorizations give the LCD directly.
Confirm both denominators divide it evenly. If the LCD is correct, dividing it by each original denominator leaves no remainder.
Method 3: the shortcut when one denominator divides the other
If the larger denominator is a multiple of the smaller one, the LCD is simply the larger denominator. No lists, no factoring. Always test for this first, because it skips a whole listing step.
Three denominators or more
The same methods apply, you just carry more numbers through. When all the denominators are different primes with no shared factors, the LCD is their plain product.
When the denominators share factors, factor them all and keep the highest power of each prime once. Powers of two collapse to the largest one, and the remaining primes appear a single time.
Rewriting fractions with the new denominator
Divide the LCD by the original denominator, then multiply the numerator and the denominator by that result. Multiplying both parts keeps the fraction's value the same while putting it over the shared bottom.
Worked examples to check against
Use these as a copy-paste reference for common pairs. Each row lists the fractions, the LCD, and the rewritten forms over that denominator.
1/2, 1/3 → LCD 6 → 3/6, 2/6 3/4, 2/5 → LCD 20 → 15/20, 8/20 1/6, 5/8 → LCD 24 → 4/24, 15/24 7/10, 1/15 → LCD 30 → 21/30, 2/30 3/8, 1/12 → LCD 24 → 9/24, 2/24 1/3, 1/4, 1/6 → LCD 12 → 4/12, 3/12, 2/12
Need one that is not on this table? Drop the fractions into our LCD calculator and it will show the rewritten forms.
When you would actually use the LCD
The LCD matters whenever fractions need a shared bottom before you can work with them. Adding or subtracting requires it, since two fractions cannot combine until they share a denominator. Comparing fractions uses it too, because once both sit over the LCD the larger numerator wins. Algebra with rational expressions follows the same logic, just with variables in the denominator.
It is not needed for multiplying or dividing fractions. If you find yourself computing an LCD to multiply, stop and see finding the product of fractions for the faster method.
Common mistakes
Multiplying the denominators when you do not have to. Two denominators where one divides the other already share a small LCD, so multiplying them out gives a needlessly large number. Always check whether the smaller denominator divides into the larger one first.
Forgetting to multiply the numerator. Changing only the bottom breaks the fraction's value. Rewriting a fraction means multiplying both the top and the bottom by the same factor.
Using the LCD when multiplying. You do not need a common denominator to multiply fractions. Multiplying straight across works fine.
The reliable habit is to check for the shortcut first, fall back to prime factorization for larger numbers, and always rewrite both the numerator and the denominator when you convert. For the underlying concept that ties this together, see what does product mean in math.
Common questions
The least common multiple is the smallest shared multiple of any set of whole numbers. The least common denominator is that same idea applied only to the denominators of fractions. The LCD of a fraction set equals the LCM of its denominators.
Use the same two methods you would for two fractions. Factor each denominator into primes, take the highest power of every prime that appears, and multiply those together. If the denominators are different primes with no shared factors, multiply them directly.
No. A common denominator is only required for adding, subtracting, or comparing fractions. To multiply, multiply the numerators together and the denominators together, then simplify.
Check whether the larger denominator is a multiple of the smaller one first. If it is, that larger denominator is the LCD. For numbers that do not share that relationship, prime factorization is faster than listing multiples.
