The number of prime numbers is infinite, so is the number of sets of co-prime numbers. Whenever the denominators are co-prime, they have no common factor except 1.
Some examples of denominators that have no common factors have been given above, you can frame an infinite number of such examples.
When we need to do an operation on two rational numbers, first we need to have a common denominator.
If the ddenominators are not the same, then we will have to multiply the denominators by each other in order to have a common denominator.
3/(x-1) + 2x/(x-3)
We notice that the denominators are different.
==> (3(x-3)/(x-1)(x-3) + 2x(x-1)/(x-1)(x-3)
Now we have the same denominator, then we will add the numerators.
Is there an infinate # of them or limited?
Let's have an example:
2/(x-1) + 1/x
We notice that the denominators are not the same.
We'll have to calculate the LCD (least common denominator)
LCD = x(x-1)
We'll multiply by LCD both fractions:
2x(x-1)/(x-1) + x(x-1)/x
Since the fractions have the same denominator, we can add them:
(2x + x - 1)/x(x-1) = (3x-1)/x(x-1)