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LCD Calculator
Find the least common denominator of fractions and rewrite each one, or find the LCD of whole numbers.
A positive fraction such as 1/4, or a whole number such as 4
A positive fraction such as 1/4, or a whole number such as 4
A positive fraction such as 1/4, or a whole number such as 4
You can enter up to 6 fractions or whole numbers.
Least common denominator is 24.
Formula
LCD = LCM of the denominators
A whole number's denominator is 1. If every entry is a whole number, the least common denominator is the LCM of those numbers.
Least common denominator (LCD)
Rewritten with denominator 24
1/4 = 6/24
1/4 = (1 × 6)/(4 × 6) = 6/24
1/6 = 4/24
1/6 = (1 × 4)/(6 × 4) = 4/24
1/8 = 3/24
1/8 = (1 × 3)/(8 × 3) = 3/24
Prime factorization
- 4 = 2²
- 6 = 2 × 3
- 8 = 2³
The least common denominator uses the highest power of each prime: 2³ × 3 = 24.
Euclidean steps
The least common denominator is the least common multiple of these denominators. For two values, lcm(a, b) = |a × b| / gcf(a, b).
First, the LCM of 4 and 6.
- 6 ÷ 4 = 1 remainder 2 (1 × 4 + 2 = 6)
- 4 ÷ 2 = 2 remainder 0 (2 × 2 + 0 = 4)
When the remainder is 0, the final divisor, 2, is the greatest common factor of 4 and 6.
lcm(4, 6) = |4 × 6| / gcf(4, 6)
|4 × 6| / 2 = 24 / 2 = 12
Fold in 8.
- 12 ÷ 8 = 1 remainder 4 (1 × 8 + 4 = 12)
- 8 ÷ 4 = 2 remainder 0 (2 × 4 + 0 = 8)
When the remainder is 0, the final divisor, 4, is the greatest common factor of 12 and 8.
lcm(12, 8) = |12 × 8| / gcf(12, 8)
|12 × 8| / 4 = 96 / 4 = 24
The least common denominator of 4, 6, and 8 is 24.
This LCD calculator finds the least common denominator, also called the lowest common denominator, for two to six fractions or whole numbers. Fractions such as 1/4, 1/6, and 1/8 are rewritten as 6/24, 4/24, and 3/24. The least common denominator is the least common multiple of the denominators, and a whole number written beside a fraction counts as a denominator of 1. When every entry is a whole number, the LCD is their least common multiple, so 4, 8, and 12 give 24. Prime factors and the Euclidean steps show why that denominator is the smallest one that works.
What is the least common denominator?
The least common denominator, also called the lowest common denominator, is the smallest positive whole number that every denominator divides evenly. It is the least common multiple of those denominators. Any larger shared number is still a common denominator, but the LCD is the least one, which keeps the renamed fractions as small as they can be.
For 1/4, 1/6, and 1/8, the denominators are 4, 6, and 8. The least common denominator is 24. The fractions are rewritten as 6/24, 4/24, and 3/24. Those three fractions are the same values as the originals, now in pieces of size 1/24, so they can be added, subtracted, or compared.
How to find the least common denominator
Factor each denominator into primes, then keep the highest power of every prime. 4 = 2², 6 = 2 × 3, and 8 = 2³. The highest power of 2 is 2³, and the highest power of 3 is 3, so the LCD is 2³ × 3 = 24. Multiply each fraction by a form of 1 that turns its denominator into 24: 1/4 × 6/6 = 6/24, 1/6 × 4/4 = 4/24, and 1/8 × 3/3 = 3/24. The renamed fractions stay over 24 even when they could reduce, because the point of the LCD is the shared denominator. 6/24 does reduce to 1/4.
Two denominators can also use the Euclidean algorithm. Divide the larger by the smaller, keep the remainder, and repeat until the remainder is 0. The last divisor is the greatest common factor, and lcm(a, b) = a × b / gcf(a, b). Fold in each extra denominator the same way. A denominator of 0 is invalid. This lesson uses positive fractions, so a minus sign is rejected. A zero numerator, such as 0/5, is allowed, and that denominator still counts. A mixed number such as 1 1/2 should be written as an improper fraction, 3/2.
The LCD of whole numbers is their LCM
A whole number written next to a fraction has a denominator of 1. Then 4 and 1/6 use the denominators 1 and 6, the least common denominator is 6, and 4 is rewritten as 4/1 = 24/6. If every entry is a whole number, there is no fraction bar to read, and the least common denominator is their least common multiple. The LCD of 4, 8, and 12 is 24. The LCM and GCF calculator works out that least common multiple, and the greatest common factor, for whole numbers on their own.
LCM of whole numbers
Frequently Asked Questions
What is the least common denominator?
The least common denominator is the smallest positive whole number that all of the denominators divide evenly. Lowest common denominator means the same thing. For 1/4, 1/6, and 1/8, the least common denominator is 24.
How do you find the least common denominator of 1/4, 1/6, and 1/8?
Take the denominators 4, 6, and 8. In primes, 4 = 2², 6 = 2 × 3, and 8 = 2³. Keep 2³ and 3, and 8 × 3 = 24. Then 1/4 = 6/24, 1/6 = 4/24, and 1/8 = 3/24.
How do you find the LCD?
Find the least common multiple of the denominators. For two denominators, lcm(a, b) = a × b / gcf(a, b). Each extra denominator is folded in with that same rule. A whole number beside a fraction counts as a denominator of 1.
Is a common denominator always the least one?
No. 48 is a common denominator of 1/4, 1/6, and 1/8, and so is 24. The least common denominator is the smallest such number, 24. A larger one still lets you add the fractions, but the LCD keeps the arithmetic smaller.
What is the LCD of the whole numbers 4, 8, and 12?
The least common denominator of whole numbers is their least common multiple. The LCD of 4, 8, and 12 is 24. The LCM and GCF calculator shows that least common multiple, with the Euclidean steps, when you are not rewriting fractions.