This process of cutting numbers down to size in order to get the pieces we want is referred to as finding the " lowest (or least) common denominator (LCD)." Yes, the "L" can stand for either "lowest" or "least," which mean the same thing. It could also stand for "littlest," we suppose, but that doesn't sound very professional.
The LCD is the number of pieces you have to reduce to so that you can easily compare your fractions.
When comparing them, the LCD is the smallest number that's a multiple of both denominators. Another way to say this is that the LCD is the smallest number divisible by both denominators. Another way to say this is... oh, you know what, you already have enough ways to say it.
To find the LCD quickly (because you never know when you'll only have 10 seconds to find one so that you can defuse a bomb in time), we use prime factorizations again.
Notice that, to find the LCD, it doesn't matter what the numerators of the fractions are. Usually, after finding the LCD, we replace both fractions with the equivalent versions whose denominator is the LCD. Having common denominators trumps having reduced fractions. If your teacher complains, you tell her we said so.