Least common denominator calculator
The least common denominator of fractions with denominators 8, 12 and 15 is 120. Eighths are scaled by 15, twelfths by 10 and fifteenths by 8 to get there.
Least common denominator
- Least common denominator The LCD is the least common multiple of the denominators.
- 120
- Each fraction over 120 Multiply the top and bottom of each fraction by the LCD divided by its denominator, which is the new numerator in each result. A fraction with a numerator other than 1 is scaled by the same factor: its numerator times that number.
- 1/8 = 15/120 1/12 = 10/120 1/15 = 8/120
- Greatest common factor
- 1
- Denominators
- 8, 12, 15
- Prime factorizations
- 8 = 2^3 12 = 2^2 x 3 15 = 3 x 5
- GCF takes the lowest power of each shared prime No factor but 1 divides all of them, though some pairs may still share one.
- no shared primes, so the GCF is 1
- LCM takes the highest power of every prime
- 2^3 x 3 x 5 = 120
- GCF is also called the greatest common divisor, highest common factor, GCD or HCF. They are all the same thing.
- Two numbers with a GCF of 1 are called coprime or relatively prime. They need not be prime themselves: 8 and 9 are coprime.
How the GCF and LCM Calculator works
The least common denominator (LCD) of a set of fractions is the least common multiple of their denominators. Enter the denominators as the numbers above and the page reports the LCD and the factor each fraction has to be multiplied by. For 3/8, 5/12 and 7/15 the denominators are 8, 12 and 15, and the LCD is 120. The numerators play no part in it.
To find it by prime factors: 8 = 2^3, 12 = 2^2 x 3 and 15 = 3 x 5. Take each prime at the highest power it reaches in any denominator (2^3, 3 and 5) and multiply: 8 x 3 x 5 = 120.
Then rewrite each fraction over 120 by multiplying its top and bottom by the same number. 120 / 8 = 15, so 3/8 = 45/120. 120 / 12 = 10, so 5/12 = 50/120. 120 / 15 = 8, so 7/15 = 56/120. Now they add like whole numbers: 45 + 50 + 56 = 151, so the sum is 151/120, or 1 31/120 as a mixed number. 151 is prime, so the fraction is already in lowest terms.
The same rewriting is how to compare fractions that do not share a denominator. 5/12 and 7/15 are hard to rank by eye; as 50/120 and 56/120, 7/15 is clearly the larger.
Multiplying the denominators together always gives a common denominator, and the method still works. It leaves bigger numbers to carry and more reducing at the end. 8 x 12 x 15 = 1,440, twelve times the LCD. With small or coprime denominators the difference is nothing; 1/3 + 1/4 has an LCD of 12 either way.
The most common mistake is not reducing first. The LCD is calculated from the denominators as written, so unreduced fractions inflate it. 6/16 and 5/12 have denominators whose LCM is 48, but 6/16 is 3/8, and 3/8 and 5/12 have an LCD of 24. Reduce each fraction before finding the LCD and the numbers stay smaller.
Mixed numbers use the denominators of their fractional parts, and a whole number counts as a denominator of 1, which never changes the LCD.
This LCD calculator handles whole-number denominators. Algebraic fractions such as 1/x + 1/(x + 1) also have a least common denominator, found the same way by factoring each denominator, but the factors are polynomials rather than primes and the calculator does not take them.
GCF and LCM Calculator FAQ
What is the least common denominator of 3/8, 5/12 and 7/15?
120, the least common multiple of 8, 12 and 15. The fractions become 45/120, 50/120 and 56/120, which add up to 151/120.
Is the least common denominator the same as the least common multiple?
It is the least common multiple of the denominators. The name changes because it is being used for fractions; the arithmetic is identical.
Can I just multiply the denominators?
Yes, that always gives a common denominator, but often not the least. 8 x 12 x 15 = 1,440, while the LCD is 120. The answer comes out the same after reducing; the arithmetic is just heavier.
Should I simplify fractions before finding the LCD?
Yes. 6/16 and 5/12 would need 48ths, but 6/16 reduces to 3/8, and 3/8 and 5/12 only need 24ths.