 In this video, let us do a rough proof to show that the product of LCM and HCF of two numbers that's called those numbers A and B is equal to the product of those numbers. So if we have two numbers A and B, we take their LCM, we take their HCF, we multiply these two, it will be the same as the multiplication of the product of the two numbers themselves A and B. So for now, let me call my HCF H and let me call my LCM L. So this should be equal to A times B. So this is what we need to prove. Now here, let us start by talking about the HCF. What is the HCF? HCF is the highest common factor. So it is a factor that is common to both A and B, right? So this means that H must divide A and H must also divide B. Let's look at an example. For example, 6 is a common factor of 12 and 42. This means that 6 divides 12 and 6 divides 42. 12 divided by 6 gives me a number 2 and 42 divided by 6 also gives me another number 7, right? So here as well, we can say that A divided by H will give me some number. Let's call that number P. And similarly, B divided by H will also give me some number. Let's call that number Q. So in this example, these numbers for us are 2 because 12 divided by 6 would give me 2 and 42 divided by 6 would give me 7. Now here comes the interesting part. Because H is the highest common factor and not just any other common factor, P and Q here will be co-primes. In other words, they will have no common factor other than 1. And we can see it from this example also. Here 2 and 7 are co-primes. There are no common factors between 2 and 7. Imagine what would happen if 2 and 7 had a common factor. What if it was not 2 and 7 but 2 and 8, for example? In that case, these two would have a common factor 2. And then 6 would not be the highest common factor in that case because then we have another factor here, 2, which can be multiplied into 6. Then the highest common factor would have become 12. But since we know that 6 is the highest common factor, we know for sure that 2 and 7 will not have any other common factors. Or in other words, they will be co-primes. So this is the property that we can use in a lot of places in our problems dealing with HCF and LCM. So definitely understand this as this is an important one. Now, coming back to the problem at hand, we have A is equal to P times H. And B is equal to Q times H. Let us now try to find the LCM of A and B. Okay, let us try to find this. Again here, I would like to take an example to make it very clear. So let's say P is again 2 and H is let's say 6. And B is 7 and H is again 6. So let's say these are the two numbers whose LCM we need to find. Lowest common multiple. So the LCM in this case needs to be a multiple of the first number. So we will have 2 into 6. But it also needs to be a multiple of the second number. Since we already have 6 here, we can just multiply it with the 7. So we have the first number here, 2 into 6. And the second number here, 6 into 7. And this is the lowest common multiple. Of course, we can multiply it by any higher number and that will still be a multiple. But this is the lowest common multiple. Similarly here, can you tell me what will be the LCM of these two numbers? Pause the video and try it yourself first. So hopefully you have tried it. P into H into Q. Right? Very similar to what we did here. Since we need a common multiple of A and B, we need a multiple of P into H. But we also need a multiple of Q into H. We already had H, so we simply multiply it with Q. This is the LCM of A and B. Now we are ready to prove our property. We have the LCM, we have the HCF. What was our HCF? The HCF was H, right? That is what we assumed. That HCF is equal to H. So if we multiply the two, LCM times HCF, what do we get? We get P times Q times H square. All right? And what about our two numbers, A and B? What do we get when we multiply them? A times B. A is P into H. So P into H and B is Q times H. So this will be Q times H. But the product of these two will be P, Q, H square. The same P, Q and H square. So we have the same products. This means that 1 is equal to 2. Or LCM into HCF is equal to A into B.