Arithmetic
Least Common Denominator
Find the smallest denominator a set of fractions can share — then see each fraction rewritten over it, ready to add or compare. Every prime-factor step shown, with a lookup chart for the pairs people search most.
Enter your fractions
Type the numerator and denominator of each fraction. We find the common denominator and rewrite them over it.
Least common denominator
12
LCD of 4, 6 = LCM of 4, 6
Built from prime powers
2² × 3 = 12
Each fraction over the LCD
- 1/4× 3/3 =3/12
- 5/6× 2/2 =10/12
Now you can add them
3/12 + 10/12 = 13/12
Updates as you type
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The basics
What is the least common denominator?
The least common denominator (LCD) is the smallest number every denominator in a set of fractions divides into evenly. It is the bridge you build before adding, subtracting, or comparing fractions — once every fraction sits over the same bottom number, the pieces are the same size and the arithmetic is easy. Crucially, the LCD is nothing new: it is just the least common multiple of the denominators.
LCD = LCM of the denominators
It is the LCM in disguise
Drop the word “denominator” and you have the least common multiple. Any rule that works for the LCM of a and b works for the LCD.
LCD of 4 and 6 = LCM(4, 6) = 12
a/b → (a·k)/(b·k)
Why fractions need it
You can only add fractions whose pieces are equal in size. Rewriting each one over the LCD makes the denominators match, so you simply add the numerators.
1/4 = 3/12 and 5/6 = 10/12, so 1/4 + 5/6 = 13/12
Method
How to find the LCD by hand
The prime-factorization route is the most reliable, especially for three or more numbers. Here it is in four steps.
- 1
Factor each denominator into primes
Break every denominator down to a product of primes. Repeated factors stay — write them as powers.
6 = 2 × 3, 8 = 2³, 12 = 2² × 3 - 2
Take the highest power of each prime
Scan across all the factorizations. For every prime that appears, keep the largest exponent you saw.
primes 2 and 3 → 2³ (from 8), 3¹ (from 6) - 3
Multiply those highest powers together
The product of the highest prime powers is the least common denominator.
2³ × 3 = 24 - 4
Rewrite each fraction over the LCD
Multiply each fraction’s top and bottom by whatever turns its denominator into the LCD. Now they share a bottom and you can add.
1/6 = 4/24, 3/8 = 9/24, 5/12 = 10/24
Worked examples
Three ways to reach the same LCD
Prime factorization, listing multiples, and the GCD formula all agree. Switch between them on the LCD of 6 and 8.
Find the LCD of 6 and 8
Factor each: 6 = 2 × 3, 8 = 2³
Take the highest power of every prime that appears.
Highest powers: 2³ × 3
LCD = 2³ × 3 = 24
Where it shows up
Why the LCD matters
A common denominator is the quiet prerequisite behind most fraction work students and tradespeople do every day.
1/4 + 5/6
Adding & subtracting
The headline use. Put both fractions over the LCD, then add or subtract the numerators while the denominator stays put.
3/8 vs 5/12
Comparing fractions
Over a shared denominator, the fraction with the bigger numerator is the bigger fraction — no decimals or cross-multiplying needed.
1/2" + 3/8"
Measurements & recipes
Mixing eighths and halves on a tape measure, or thirds and quarters in a recipe, means finding a common denominator first.
Untangling terms
LCD, LCM, and GCD
Three closely related ideas that get mixed up. They pull in opposite directions, and two of them are the same thing.
LCD
Least common denominator
The smallest shared multiple of a set of denominators. A fractions-flavored name for the LCM.
LCD of 4, 6 = 12
LCM
Least common multiple
The smallest number all the inputs divide into. Identical to the LCD mathematically; use the LCM calculator when the inputs are whole numbers, schedules, or cycles instead of fraction denominators.
LCM of 4, 6 = 12
GCD
Greatest common divisor
The largest number that divides every input. The opposite end — and the shortcut that makes the LCD easy: LCD = product ÷ GCD.
GCD of 4, 6 = 2
Worth knowing
Rules that speed things up
A few shortcuts let you read off the LCD without grinding through full factorizations.
coprime → a × b
No common factor? Multiply
When two numbers share no common factor, the LCD is simply their product. This is why any two different primes multiply straight to their LCD.
b ÷ a is whole → b
One divides the other
If the larger denominator is a multiple of the smaller, the larger one is already the LCD. The smaller adds nothing new.
LCD ≥ largest input
It is never smaller than the biggest
The LCD has to be divisible by every denominator, so it can never be less than the largest number in the set.
(a × b) ÷ GCD
The two-number shortcut
For a pair, the LCD equals their product divided by their greatest common divisor — usually the fastest hand method.
chain pairwise
More than two? Chain it
Find the LCD of the first two, then the LCD of that result with the third, and so on. The order never changes the answer.
any common denom works
Least just keeps it tidy
Using a non-least common denominator still gives a correct sum — you simply have larger numbers to simplify at the end.
Quick chart
What is the LCD of two numbers?
Type any two numbers, or tap a common pair. Every tile is the least common denominator of the two numbers shown — the answers people search for most. The most-asked pairs have a full worked page of their own, including the LCD of 4 and 8, 7 and 8, and the three-number case, the LCD of 3, 4 and 5.
&
LCD
12
(4 × 6) ÷ GCD(4, 6) = 24 ÷ 2 = 12
LCD of 4 & 88LCD of 7 & 856LCD of 5 & 735LCD of 3 & 824LCD of 6 & 742LCD of 7 & 963LCD of 8 & 1224LCD of 6 & 824LCD of 5 & 840LCD of 8 & 1040LCD of 3 & 515LCD of 4 & 936LCD of 3 & 412LCD of 9 & 1236LCD of 6 & 918LCD of 4 & 612LCD of 4 & 520LCD of 2 & 88LCD of 5 & 630LCD of 4 & 728LCD of 2 & 44LCD of 4 & 1020LCD of 9 & 1545LCD of 3 & 721LCD of 3 & 66LCD of 10 & 1530LCD of 8 & 972LCD of 2 & 36LCD of 3 & 99LCD of 5 & 945LCD of 11 & 12132LCD of 5 & 1010
Blue tiles open a full worked answer page for that pair; the rest load into the boxes above.
Common mistakes
Where people slip up
The LCD itself is rarely the hard part — the errors hide in what you do with it.
forgot the numerator
Scaling only the bottom
When you multiply a denominator to reach the LCD, you must multiply the numerator by the same factor. Change one without the other and you have changed the value of the fraction.
added denominators
Adding the denominators too
Over a common denominator you add the numerators only; the shared denominator stays exactly as it is. It does not double.
just multiplied
Multiplying when a smaller LCD exists
Multiplying all the denominators always gives a common denominator, but rarely the least. You will be left simplifying a needlessly big fraction.
skipped the GCD
Ignoring shared factors
Treating 8 and 12 as coprime gives 96 instead of 24. Always check for a common factor before you reach for the product.
FAQ
Common questions
The least common denominator (LCD) is the smallest positive whole number that every denominator in a set of fractions divides into evenly. It is the common denominator you rewrite fractions over before adding, subtracting, or comparing them.
Yes. The least common denominator is just the least common multiple (LCM) applied to the denominators of fractions. Computing the LCD of 4 and 6 is identical to computing the LCM of 4 and 6 — both give 12.
Three common ways: list the multiples of each denominator and take the first one they share; factor each denominator into primes and multiply the highest power of every prime; or use LCD(a, b) = (a × b) ÷ GCD(a, b) and chain it across more than two numbers.
The LCD of 6 and 8 is 24. Since 6 = 2 × 3 and 8 = 2³, the LCD takes 2³ × 3 = 24 — the smallest number both 6 and 8 divide into.
The LCD of 3, 4, and 5 is 60. The three numbers share no common factors, so the LCD is simply their product, 3 × 4 × 5 = 60.
Fractions can only be added or subtracted when the pieces are the same size — that is, when the denominators match. Rewriting each fraction over the LCD makes every piece the same size, so you can add the numerators directly.
No — any common denominator works, and the product of all denominators is always one. But the least common denominator keeps the numbers smallest and usually means less simplifying at the end, so it is the preferred choice.
When two numbers share no common factor (including any two different primes), their LCD is just their product. The LCD of 5 and 7 is 35, and the LCD of 3 and 11 is 33.
Keep going
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Author
Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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