There is no such thing as a greatest common denominator. For example, the LEAST common denominator of 3 and 4 is 12, but any multiple of 12 will also be a common denominator.
For very small numbers, the fastest way to find the LEAST common denominator (or least common multiple) is to try out multiples of the larger number. For example, to get the least common multiple of 6 and 4: 6 is not a multiple of 4, but the next multiple of 6 (6 x 2 = 12) is, so that's the least common multiple.
For numbers of intermediate size, you may want to use prime factorization. That's the method common taught in schools. To repeat with the numbers above: 6 = 3 x 2; 4 = 2 x 2. Gathering all the prime factors, and using duplicates (those that appear in both numbers) only once (or using the highest power that appears in any of the numbers, you get 2 x 2 x 3 = 12.
For fairly large numbers, use Euclid's algorithm to find the greatest common factor first. In this case, 6 divided by 4 gives a remainder of 2, and 4 divide by 2 gives a remainder of 0. The last non-zero number in this sequence of divisions is 2, so that's the greatest common factor.
Once you have the greatest common factor, use the property that a x b = lcm(a, b) x gcf(a, b). In other words, the least common multiple is equal to a x b / gcf(a, b), in this example, 6 x 4 / 2 = 12.
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Example: 30 and 42
Factor them.
2 x 3 x 5 = 30
2 x 3 x 7 = 42
Select the common factors.
2 x 3 = 6, the GCF
The fastest and easiest way to find a least common denominator between two fractions is to compare the prime factorizations of the denominators in question.
Finding the GCF of the numerator and the denominator of a fraction and dividing them both by it will give you the simplest form of that fraction. Finding the LCM of unlike denominators and converting them to it will make it possible to add and subtract unlike fractions.
I'm not sure finding a common numerator will help you very much. The common denominator of those two numbers is 420.
Finding the GCF will help you when you are trying to reduce fractions.
If you take all the common prime factors between numbers and multiply them it will give you the gcf.
When adding and subtracting unlike fractions, it is necessary to find a least common denominator. It's the same process as finding an LCM. You can simplify a fraction by finding the GCF of the numerator and denominator and dividing them both by it.