Find the least common denominator of any set of fractions in seconds — with prime-factorization steps shown. Free, fast, and mobile-friendly.
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Adding fractions with different denominators is impossible until the fractions share one — and the smallest denominator they can all share is the least common denominator (LCD). Our free LCD Calculator finds it instantly for two or three denominators, showing the prime factorization of each so you can follow the method, not just copy the answer.
The LCD is simply the least common multiple (LCM) of the denominators. For 1/4 + 1/6, the denominators 4 and 6 have multiples 4, 8, 12, 16… and 6, 12, 18… — the smallest shared multiple is 12, so 12 is the LCD. Listing multiples works for small numbers, but for larger denominators the prime-factorization method is far more reliable, and it is exactly what the calculator demonstrates step by step.
Whether you are helping with homework, scaling recipes, combining measurements in a workshop, or working through algebra, the LCD is the gateway step: once every fraction is rewritten with the LCD, numerators add directly and the problem collapses into ordinary arithmetic.
The least common denominator of a set of fractions is the smallest positive number that is a multiple of every denominator. It matters because fractions can only be added or subtracted when they are expressed in the same units — just as you cannot add 3 inches to 2 centimeters without converting, you cannot add 1/4 to 1/6 until both are expressed in twelfths (or another common denominator). The LCD is the least such denominator, which keeps the numbers as small as possible.
Any common multiple would technically work: 1/4 + 1/6 could be computed with denominator 24 or 48. But larger denominators mean uglier intermediate fractions and a harder final simplification. The LCD minimizes the arithmetic, which is why teachers insist on it. There is a deep reason it is also the LCM: the set of common denominators of fractions with denominators d1, d2 is exactly the set of common multiples of d1 and d2, so the least of one is the least of the other.
The prime-factorization method finds the LCM without listing endless multiples. Factor each denominator into primes, then for each prime take the highest power appearing in any factorization, and multiply those together. For 12 = 2² × 3 and 18 = 2 × 3², the highest powers are 2² and 3², giving 4 × 9 = 36. Listing multiples of 12 and 18 would eventually reach 36 too, but factorization gets there in two lines.
The LCD also appears when comparing and ordering fractions: rewriting 5/12, 7/18, and 11/24 with LCD 72 gives 30/72, 28/72, 44/72, making the order obvious. And in algebra, clearing denominators in an equation — multiplying both sides by the LCD — is the standard first move for solving rational equations.
LCD(d₁, d₂, …) = LCM(d₁, d₂, …) | LCM(a, b) = |a × b| / GCD(a, b)
Equivalently: factor each denominator into primes, take the highest power of each prime, and multiply. For three or more numbers, apply the LCM pairwise: LCM(a, b, c) = LCM(LCM(a, b), c).
Factor: 4 = 2², 6 = 2 × 3. Highest powers: 2² and 3¹. LCD = 4 × 3 = 12. Check: multiples of 4 are 4, 8, 12…; multiples of 6 are 6, 12…; the first shared one is 12. So 1/4 + 1/6 = 3/12 + 2/12 = 5/12.
Factor: 8 = 2³, 12 = 2² × 3. Highest powers: 2³ and 3. LCD = 8 × 3 = 24. Then 3/8 + 5/12 = 9/24 + 10/24 = 19/24. Notice the LCD (24) is larger than either denominator but smaller than their product (96) — the GCD of 8 is what the product method would waste.
Factor: 6 = 2 × 3, 10 = 2 × 5, 15 = 3 × 5. Highest powers: 2, 3, 5. LCD = 2 × 3 × 5 = 30. So 1/6 + 1/10 + 1/15 = 5/30 + 3/30 + 2/30 = 10/30 = 1/3. Pairwise check: LCM(6, 10) = 30, LCM(30, 15) = 30. Consistent.
When denominators share no prime factors, the LCD is simply their product. 7 is prime; 9 = 3². Highest powers: 7 and 3², so LCD = 63. Then 2/7 + 4/9 = 18/63 + 28/63 = 46/63. In general, if GCD(a, b) = 1, then LCM(a, b) = a × b.
A recipe calls for 1/4 cup of sugar and 1/3 cup of honey, and you want the total sweetener. The LCD of 4 and 3 is 12: 1/4 = 3/12 and 1/3 = 4/12, so the total is 7/12 cup. Without the LCD you would be stuck guessing; with it, the addition is trivial. Scaling the recipe up or down uses the same denominator throughout.
A woodworker needs to combine a 5/8-inch board and a 7/12-inch board to check total thickness. LCD(8, 12) = 24: 5/8 = 15/24 and 7/12 = 14/24, totaling 29/24 inches, or 1 5/24 inches. Tape measures are marked in 16ths, so she might further convert, but the LCD step is what makes the addition exact rather than eyeballed.
In personal finance, suppose you earn 1/6 of a project fee in January and 1/4 in February and want the fraction completed. LCD(6, 4) = 12, giving 2/12 + 3/12 = 5/12 of the project done. Anytime quantities arrive in different fractional chunks — time sheets, ingredient lists, construction materials — the LCD is the tool that puts them on a common footing.
The prime-factorization method is not a trick — it follows directly from the Fundamental Theorem of Arithmetic, which says every integer factors uniquely into primes. A number M is a multiple of d exactly when M’s prime factorization contains every prime of d’s factorization at an equal or higher power. So a common multiple of several denominators must contain, for each prime, at least the highest power appearing anywhere — and the least such number contains exactly those highest powers and nothing else.
This also explains the formula LCM(a, b) = a×b / GCD(a, b). The product a×b contains every prime at the sum of its powers in a and b; dividing by the GCD (which holds the minimum powers) leaves the maximum powers — precisely the LCM. The calculator uses this GCD-based route internally because Euclid’s algorithm computes GCDs in microseconds, then displays the prime factorizations so you can see the classical method too.
For three or more denominators, the pairwise reduction LCM(a, b, c) = LCM(LCM(a, b), c) works because “being a multiple of all three” is the same as “being a multiple of the least common multiple of the first two, and of the third.” The calculator chains this automatically when you fill in the third box.
Factor: 9 = 3², 12 = 2² × 3. Highest powers: 2², 3². LCD = 4 × 9 = 36. Then 5/9 − 1/12 = 20/36 − 3/36 = 17/36. Subtraction needs the LCD exactly as much as addition does.
Factor: 14 = 2 × 7, 35 = 5 × 7. Highest powers: 2, 5, 7. LCD = 70. Via the GCD route: GCD(14, 35) = 7, so LCM = 14×35/7 = 70. Then 3/14 + 2/35 = 15/70 + 4/70 = 19/70.
A handy check: the LCD of a set of denominators always divides their product, and it equals the product exactly when the denominators are pairwise coprime. If your computed LCD does not divide evenly into the product of the denominators, something went wrong — recompute. And if one denominator already divides another (like 4 and 12), the LCD is simply the larger one.
Find the LCD of 10 and 15, then compute 3/10 + 2/15. (Hint: 10 = 2 × 5, 15 = 3 × 5.)
A tank is 5/12 full and you add 1/8 of a tank. What fraction is full now? (Hint: LCD(12, 8) = ?)
Three friends split a bill unevenly: Ana pays 1/4, Ben pays 1/6, and Cara pays the rest. What fraction does Cara pay? The LCD of 4 and 6 is 12: Ana’s share is 3/12 and Ben’s is 2/12, totaling 5/12, so Cara pays 7/12. One LCD computation turns a confusing “who owes what” moment into a single subtraction.
Add a fourth friend paying 1/3 and the LCD of 4, 6, 3 is still 12 (since 3 divides 12): shares become 3/12 + 2/12 + 4/12 = 9/12, leaving 3/12 = 1/4 for the last person. Notice how recognizing that 3 already divides the LCD saved a recomputation — a pattern worth spotting in timed tests.
The LCD Calculator finds the least common denominator of two or three denominators by computing their least common multiple (via GCD), displaying each denominator’s prime factorization and the two-step LCM construction, so users can rewrite fractions on a common denominator for addition, subtraction, or comparison.
The least common denominator (LCD) of a set of fractions is the smallest positive number that is a multiple of every denominator. It lets you rewrite all the fractions with the same denominator so their numerators can be added or subtracted directly.
Factor each: 4 = 2², 6 = 2 × 3. Take the highest power of each prime (2² and 3) and multiply: 4 × 3 = 12. So the LCD of 4 and 6 is 12.
Yes — the LCD of a set of fractions is exactly the least common multiple (LCM) of their denominators. “LCD” is the name used when the numbers are denominators; “LCM” is the general term.
The product is always a common denominator, but usually not the least one. For 4 and 6 it gives 24 instead of 12, doubling your arithmetic. Only when the denominators share no factors (are coprime) does the product equal the LCD.
Apply the LCM pairwise: first find LCM of the first two denominators, then find the LCM of that result with the third. Equivalently, factor all three, take the highest power of each prime across all factorizations, and multiply.
Factor: 8 = 2³, 12 = 2² × 3. Highest powers: 2³ and 3¹, so LCD = 8 × 3 = 24. Check: 24 ÷ 8 = 3 and 24 ÷ 12 = 2, both whole numbers.
A fraction’s denominator names the size of the pieces. You can only combine counts of same-sized pieces — 3 twelfths plus 2 twelfths is 5 twelfths — so unlike denominators must first be converted to a shared one.
Rewrite each fraction with the LCD: multiply its numerator and denominator by whatever turns the old denominator into the LCD. Then add or subtract the numerators, keep the LCD, and simplify the result if possible.
Cite this page as: LCD Calculator. Sigma Calculator. Retrieved October 9, 2026, from https://sigmacalculator.net/math-calculators/lcd-calculator/
Last updated: October 9, 2026