Find the least common denominator of each set of fractions. 2 8 , 9 15 y , 3 20 \frac28,\frac{9}{15y},\frac{3}{20}

This problem asks us to find the least common denominator of the fractions that were given. And notice we have 2/8, 9/15 Y, and 3/20. Now, to solve this problem, I can recognize something. Finding the least common denominator is finding the least common multiple of the denominators. Now, I noticed for my denominators here, one of them has a Y. And whenever we have variables in our denominators when finding the LCD, it's typically best to use prime factorization to solve the problem. So, let's go ahead and do that. So, my denominators are 8, 15 Y. And 20 Now, I need to go ahead and break these down using prime factorization and use this to find the LCD. Now, I'm gonna go ahead and start with 88 breaks down into 2 times 2 times 2. Now, let's move to 15 Y. 15 Y breaks down into 3 times 5, that gives me 15, and then this Y is just a prime factor. Now, from here, we have 20, and 20 breaks down into 2 times 2 times 5. And that is everything broken down into prime factors, cause remember, we treat variables as just prime factors alone. Now, let's go ahead and see if we can pick out the unique prime factors. And this is where things get a bit tricky cause you have to be careful here. So, starting with this 8, I can see we have 2 x 2 x 2. Those are all of my unique prime factors. And then I can see we have this 3 for the 15 Y, we have this 5, and we have this Y. Now, what about for the 20? Well, I can see we have 2 x 2, but we already had 2 x 2 x 2 up here. So, we do not need to include these 2s, they are not unique. I can also see we have this 5, but we already had a 5 for the 15Y. So, this 5 is not unique either. So, the unique prime factors are only going to be these ones we saw for the 8 and the 15Y. So, in order to find our LCD, well, this is just going to be 2 x 2 x 2 x 3 x 5 x Y, a product of all the unique prime factors. And multiplying everything out here, you should end up with 120 Y. So, 120 Y is the LCD and the solution to this problem.