 In this video, let us try and find the greatest common divisor of some decimal numbers. 2.16, 0.72, 1.8. To do this, what we can do is we can get a common number out of all of these decimals. Let me show you how. So 2.16 can be written as 2.16 by 100. 0.72 can be written as 7.2 by 100 and 1.8 can be written as 1.8 by 10. But we want to get a common number out. So let me make it 100 by multiplying with 10 in the numerator and denominator of this last term. Now we can see that we have all the denominators as 100 and we can get the 1 by 100 out of this GCD. Yes, we can. This is a property of GCDs. Let me just write it on the right hand side. That if we have two numbers A and B and we multiply them with some common number K, then the GCD of that new number will be K times the GCD of the original numbers A and B. So we can get a common number K out of GCDs just like we do. Just like we distribute K in multiplication, we can also distribute K in GCDs. How amazing is that? I will show you a proof of this towards the end of the video. But first let us finish this by calculating the GCD of these three numbers. So let us see what we have here. I want to use prime factorization for this. 2 1 6 is I know 6 cubed. So this is 2 cubed into 3 cubed because 6 is 2 into 3. Then 72 I know is 8 times 9. 8 is 2 cubed and 9 is 3 squared. And 180 is 18 into 10. 10 is 2 into 5 and 18 is 2 into 9. So another 2 and 9 is 3 squared. So I have written the prime factors. Now I want to find the common factors. So 2 square is common and 3 square is also common. 3 cubed is not 5 is not common. So only 2 square and 3 square are the common factors here. So this will be 2 squared into 3 squared and we are done. So this is 4 into 9 36 36 by 100 and the answer is 0.36. And just like that, we have found the GCD of these decimal numbers. So whenever you are finding GCD, just get these decimals to have a common denominator and get that common denominator out and you can find the GCD. Now for the second part, I want to just show you the proof of this video for those of you who are interested. A rough proof. So let's say we have two numbers A and B whose GCD is H. What does this mean? This means that H divides A and H divides B. So A by H will be some number P and B by H will be some number Q. And what will be the property of P and Q? We have used this property before. The property of P and Q will be that they are co-primes. That they will have no other factor in common because if they did then A and B would have a higher factor than H in common. But that is not the case. H is the highest common factor. So therefore P and Q are co-primes. Now what does this mean? This means that A is equal to P into H and B is equal to Q into H. Now let us multiply K on both sides. This means that K A is equal to P into K H and K B is equal to Q into K H. Okay, I think you can see where I'm going with this. So now we have two numbers K A and K B and P and Q are multiplied with K H where P and Q are co-primes. What does this mean? This means that K H, K H is the GCD of K A and K B, right? This is what it means because K H divides K A and we get P and K B divides K H and we get Q and P and Q are co-primes. So we have just used the reverse of the property. K H is the GCD of K and K B. That means, what does that mean? That means GCD of K A and K B is K times H, but what was H? H was the GCD of A and B. So this is how we can roughly prove this property and this is the property that we used. We can also just see this property from intuition, right? If I multiply all of these numbers with some new common factor 7, this is our K, then this will definitely go into my GCD. So if I multiply all the numbers with some common factor, then that will definitely be multiplied with the GCD. We could have seen it this way as well.