LCD — Least Common Denominator
Find the Least Common Denominator (LCD) of up to four denominators, with prime factorization method shown step-by-step.
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How to use this calculator
The LCD of a set of fractions is the Least Common Multiple of their denominators. Found by taking each prime factor at its highest power across all denominators.
- 1
Enter 2–4 denominators (leave unused fields at 0).
- 2
The LCD is calculated and the prime factorization method is shown.
- 3
The multiplier for each denominator (lcd/d) is also shown — multiply both numerator and denominator by this to convert to the LCD.
Frequently asked questions
What is the difference between the LCD and the LCM?
They are the same number. The least common denominator of a set of fractions is the least common multiple of their denominators. For denominators 4, 6 and 10 it is 60. The name changes only because you are using it to rewrite fractions with a shared bottom number.
What counts as a valid denominator?
A positive whole number. Decimals such as 2.5 and negatives such as −4 return an error telling you to use positive whole numbers. A field left at 0 is treated as unused, so you can enter two, three or four denominators. Entering only one returns “Enter at least two denominators”.
How do I use the multipliers?
Multiply both the numerator and denominator of a fraction by its multiplier to rewrite it over the LCD. With denominators 4, 6 and 10 the LCD is 60 and the multipliers are 15, 10 and 6. So 1/4 + 1/6 + 1/10 becomes 15/60 + 10/60 + 6/60 = 31/60.
Does it matter if a fraction can be simplified?
The calculator sees denominators only. If you enter 6 for a fraction like 3/6, it uses 6, even though the fraction equals 1/2. Simplify first when you want the smallest possible denominator.
What if I enter the same denominator twice?
It works, and the output lists each entry separately. Entering 6 and 6 gives an LCD of 6 with a multiplier of 1 for both. Entering 1 and 1 gives an LCD of 1, and the formula shows 1.
Is there a size limit?
The answer must be below 9,007,199,254,740,991, the largest integer a standard JavaScript number can hold exactly. Beyond that, the calculator returns “Result is too large to calculate exactly” rather than a rounded guess. Denominators of 1,000,000,000 and 999,999,937 trip this limit.
Can it add the fractions for me?
No. It finds the common denominator and the multipliers, which are the part people most often get wrong. The numerators are yours to multiply and add, as in the 31/60 example above.
Least common denominator with prime factorization
How the LCD is built
Factor each denominator into primes, then take every prime that appears and keep the highest power it reaches in any denominator. For 4, 6 and 10 the factorizations are 2², 2 × 3 and 2 × 5, so the LCD is 2² × 3 × 5 = 60. The calculator displays that formula, each factorization, and the multiplier for each denominator, which is the LCD divided by it. Multiplying the denominators together would give a common denominator too, but usually a much larger one.
| Denominators | LCD | Prime formula | Multipliers |
|---|---|---|---|
| 4, 6, 10 | 60 | 2² × 3 × 5 | 15, 10, 6 |
| 12, 18 | 36 | 2² × 3² | 3, 2 |
| 3, 5, 7 | 105 | 3 × 5 × 7 | 35, 21, 15 |
| 4, 6, 8, 9 | 72 | 2³ × 3² | 18, 12, 9, 8 |
| 6, 6 | 6 | 2 × 3 | 1, 1 |
Worked example: three fractions
To add 1/4 + 1/6 + 1/10, enter 4, 6 and 10. The LCD is 60 and the multipliers are 15, 10 and 6. Rewritten, the fractions are 15/60, 10/60 and 6/60, which add to 31/60. Without the LCD, a product of the denominators would be 240, giving the correct sum only after more simplifying. The calculator shows the multipliers because they are the step that is easy to get wrong.
Worked example: shared factors save work
Denominators 12 and 18 multiply to 216, yet the LCD is just 36, because both share a factor of 6. The prime formula 2² × 3² makes this visible: 12 contributes 2² and 18 contributes 3². Whenever denominators share factors, expect the LCD to be far below their product. When denominators share no factors, as with 3, 5 and 7, the LCD is just the product, 105.
Four denominators at once
With 4, 6, 8 and 9 the LCD is 72 and the multipliers are 18, 12, 9 and 8. The power of 2 is controlled by 8 (2³) and the power of 3 by 9 (3²), and the smaller powers in 4 and 6 are absorbed. This is a useful check for mixed schedules, gear teeth and repeating cycles, where the LCD is also the point at which all cycles line up again.
Input rules and limits
Use positive whole numbers; leave unused fields at 0. Decimals and negatives are rejected instead of being rounded silently, and results beyond about nine quadrillion are refused because they could not be represented exactly. The calculator does not reduce your fractions or perform the addition itself.
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