 In the last video, we talked about how to find the greatest common divisors of two fractions. We saw how to do this the brute way. We listed down all the divisors of 1 by 4 and 1 by 2 and saw which was the common one among them and which was the greatest one among those common ones. This gives us the greatest common divisors. But in this video, I want to tell you about a trick to do this. Not doing it the brute way, not listing down all the divisors and then finding the greatest common one. But a trick on how to do this for fractions. Let's see. The trick is, if we want to find the greatest common divisors of any number of fractions. For now, let's take two fractions a, b and a by b and c by d. Then this will be equal to the greatest common divisors of a, c which is the numerators divided by the LCM and let me write this down in a different color so that you know upstairs we have gcd but downstairs we have the LCM of b and d. So the greatest common divisors of these two fractions will be the greatest common divisor of the numerators divided by the least common multiple of the denominators. I won't prove this for you but what I can do is I can verify it for these numbers. In the last video, we showed that this should be equal to 1 by 4. Let's try using this formula. So we want to find the gcd of 1 by 2 and 1 by 4. According to this formula, this should be equal to the gcd of 1, 1 divided by the LCM of 2 and 4. So gcd of 1, 1 is 1 and what is the least common multiple of 2 and 4. So 4 into 1 is 4 and 2 into 2 is 4. So 4 is the least common multiple. Yes. Fair. This works. Now let's try it for a different set of numbers. Let's try and find the gcd of 15 by 4 and 9 by 14. This seems fairly tedious to do without this trick. So why don't you give it a short first, use this trick and try to find the answer and then we'll do it together. Okay, hopefully you have tried this. So this will be equal to the gcd of 15 and 9 divided by the LCM of 4 and 14. Okay, so what does this give me? gcd of 15 and 9. This divides by 3. This also divides by 3. Is there any other common divisor? 6, 7, 8, 9. Okay, so none of them. So the greatest common divisor is 3. And downstairs we'll have the LCM of 4 and 14. So 4 is 2 to the power 2 and 14 is 2 into 7. So the common multiple will be 2 into 7, but we need another 2 to get 4. So it will be 2 square into 7, that is 28. This is the gcd of these two numbers. To confirm that this is a divisor for these two fractions, why don't you divide this by this and the second fraction by this and see that you're getting integers. Alright, I hope we're clear with this trick, but I want to lay out a word of caution here. This trick works if these fractions are in their simplest forms. That means there are no cancellations remaining. We cannot simplify A by B further. Let me show you with an example. Let's say we are finding the gcd of 10 by 7 and 3 by 9. Then will this be equal to the gcd of 10 and 3 divided by the LCM of 7 and 9? Will this be equal? The answer is no, it won't. Because 3 by 9 is not in its simplest form. We must bring it in its simplest form to 1 by 3 and then we can use this formula. So keep this in mind while using this trick. Now finally, I also want to mention that this trick works for even more than two fractions. So let's get this away. And if we want to find the gcd of three fractions, let's say 18 by 5, 6 by 5 and 7 by 10, then this will be equal to the gcd of 18, 6 and 7 divided by the LCM of 5, 5 and 10. Just to make sure there is no simplification remaining, 6 by 5, 18 by 5, yeah. So this also works. And another interesting fact, final interesting fact before I end this video is that if we wanted to find the LCM of these three fractions or any three fractions for this matter, the formula would just reverse. So we would have LCM on top, 18, 6 and 7 and we would have gcd on the bottom. Again let's use a different color, gcd of 5, 5 and 10. So I'm not going to solve this for you, but just wanted to let you know all these formulas that we can use to find the gcd's and LCM's of fractions.