Quick answer
The LCD (Least Common Denominator) is the smallest number all denominators divide into evenly — the LCM of the denominators. For 1/4 and 1/6: LCD(4,6) = 12. Formula: LCD = (a × b) ÷ GCD(a, b).
How to use this calculator
Enter the denominators of two or three fractions (the denominator values only — not the full fractions). The calculator returns the LCD, the prime factorisation of each denominator, and shows each denominator scaled to an equivalent fraction with the LCD.
What is the LCD?
The Least Common Denominator (LCD) is the smallest positive integer that each of the given denominators divides into without a remainder. It is identical to the LCM (Lowest Common Multiple) of those denominators. The LCD is used to convert fractions to a common denominator so they can be added, subtracted, or compared directly.
LCD formula
The fastest two-denominator method uses the GCD:
$$\text{LCD}(b, d) = \text{LCM}(b, d) = \frac{|b \times d|}{\gcd(b, d)}$$
$$\text{LCD}(4, 6) = \frac{4 \times 6}{\gcd(4,6)} = \frac{24}{2} = 12$$
GCD(4,6) = 2. LCD = (4 × 6) ÷ 2 = 12. Verify: 4 divides 12 (12/4 = 3 ✓) and 6 divides 12 (12/6 = 2 ✓).
$$\text{LCD}(3, 4) = \frac{3 \times 4}{\gcd(3,4)} = \frac{12}{1} = 12$$
GCD(3,4) = 1 (coprime), so LCD = 3 × 4 = 12.
Prime factorisation method
Factorise each denominator, then take the highest power of each prime factor across all factorisations:
$$\text{LCD}(12, 18): \quad 12 = 2^2 \times 3, \quad 18 = 2 \times 3^2 \implies \text{LCD} = 2^2 \times 3^2 = 36$$
12 = 2² × 3 and 18 = 2 × 3². Highest powers: 2² and 3². LCD = 4 × 9 = 36. Verify: 12 | 36 ✓ and 18 | 36 ✓.
Three or more denominators
Extend by computing LCM pairwise:
$$\text{LCD}(4, 6, 10) = \text{LCM}(4, \text{LCM}(6, 10)) = \text{LCM}(4, 30) = 60$$
First, LCM(6,10): GCD(6,10) = 2, LCM = 60/2 = 30. Then LCM(4,30): GCD(4,30) = 2, LCM = 120/2 = 60. LCD = 60.
| Denominators | GCD | LCD | Equivalent fractions |
|---|---|---|---|
| 2, 3 | 1 | 6 | 1/2 → 3/6, 1/3 → 2/6 |
| 4, 6 | 2 | 12 | 1/4 → 3/12, 1/6 → 2/12 |
| 3, 5 | 1 | 15 | 2/3 → 10/15, 3/5 → 9/15 |
| 8, 12 | 4 | 24 | 5/8 → 15/24, 7/12 → 14/24 |
| 6, 9 | 3 | 18 | 5/6 → 15/18, 4/9 → 8/18 |
| 4, 5, 6 | — | 60 | 1/4→15/60, 1/5→12/60, 1/6→10/60 |
Why the LCD matters
Adding 1/4 + 1/6 without a common denominator is impossible directly. Using the product (24) also works: 6/24 + 4/24 = 10/24 = 5/12. But using the LCD (12) keeps numbers smaller: 3/12 + 2/12 = 5/12 — no simplification needed. The LCD minimises the arithmetic involved and reduces the chance of error.
The LCD is also the right denominator to use when comparing fractions: converting all fractions to the LCD makes comparison by numerator immediate.
Common mistakes
Using the product instead of the LCD
For 3/4 + 1/6, using 4 × 6 = 24 gives 18/24 + 4/24 = 22/24, which simplifies to 11/12. Correct, but more work. LCD = 12 gives 9/12 + 2/12 = 11/12 directly — no simplification needed.
Confusing LCD with GCD
GCD is the largest number that divides both values. LCD is the smallest number that both values divide into. GCD × LCM = product of the two numbers. They are related but opposite concepts.
Frequently asked questions
What is the LCD (Least Common Denominator)?
The smallest number that all denominators divide into evenly. It equals the LCM of the denominators.
How do you find the LCD?
Use LCD = (a × b) ÷ GCD(a, b), or factorise each denominator and take the highest power of each prime.
What is the LCD of 1/4 and 1/6?
12. GCD(4,6) = 2, so LCD = (4×6)/2 = 12. Convert: 3/12 and 2/12.
Is the LCD the same as the LCM?
Yes. LCD of fractions = LCM of their denominators. The terms are used interchangeably.