Harvard interview question that will test your math skills

Let's find the values of x and y. Welcome back to my channel. This question seems to be quite difficult but it's actually very easy. Yes, this is how we are going to go about it. First of all, you remember that in algebra identity, we have that a squared minus b squared is equal to a minus b multiplied by a plus b. Therefore, this is going to form the foundation of our solution because that way we will be able to express this in this form and then we find the value of x and y. Welcome back to my channel. If this is the first time you are tuning in, we drop the red carpet for you. Let's start. Like I said, we are going to express this in this form x minus y multiplied by x plus y is equal to 36. Now, we need to find the factor pairs of 36. The factor pairs of 36 is 36 multiplied by 1. Then, we have 9 multiplied by 4. We also have 12 multiplied by 3. We have 6 multiplied by 6 and we have 18 multiplied by 2. These are the factor pairs of 36. Now, we can also say that 1 multiplied by 36 will go as well. 4 multiplied by 9 can go as well. 3 multiplied by 12 can go as well. 6 multiplied by 6 is the same. 2 multiplied by 18 is also correct. Now, when we look at what we have on the board, we have that x minus y multiplied by x plus y is equal to 36. Now, if you take away y from a number and you add y to that same number, definitely x plus y must be greater than x minus y, right? So, we can write that x minus y is less than x plus y. Now, because we have these, that will help us to narrow down these factors and help us to pick the ones that might fit into this equation. If this is the first time you are coming across this channel, please click the subscription button and turn on your notification bell so that you don't miss our amazing videos. Always checking out on us. We always have new products to push out. Thank you very much. Now, if you look at what we have on the board, since x minus y must be less than x plus y, that simply means that we cannot pick this because here x minus y is greater. All the ones in our right-hand side, we are going to cancel all of them. So, that's where we are cleaning this space and we are using these factors. Okay. Now, the next obstacle that we have is being able to narrow down these factor pairs to pick the ones that actually satisfy this equation. Please don't forget to watch to the end because we are going to prove that the answer we got is correct. Okay. So, you wouldn't like to miss that. Now, looking at this, let's pick 1 times 36. Okay. So, we are going to say that x minus y multiplied by x plus y is equal to 1 multiplied by 36. If we make that statement, it simply means that x minus y will give us 1 and x plus y will give us 36. Okay. So, if we decide to add these two equations together, we have x plus x is 2x and minus y plus plus y is 0, right? And here, we have 1 plus 36 is 37. And then, if we decide to divide both sides by 2, we are going to have x to be a fraction. So, from here, you can see that our x is 37 divided by 2 and this is not a natural number and we need our x to be a natural number. Okay. So, that means that this is rejected. Now, let's check the second one. If we check the second set, which is, oops, sorry, 1 times 36, 4 times 9, 3 times 12, 6 times 6, 2 times 18. Okay. We are going to eliminate 6 times 6 as well because you can see that both of them are equal and we've already been told that x minus y must be less than x plus y. So, we're also eliminating 1 times 36 because x gives us a number that is not a natural number. Alright. Now, let's go to 4 times 9. For 4 times 9, we have x minus y multiplied by x plus y is equal to 4 multiplied by 9. So, moving forward, we have that x minus y will be equal to 4, right? And x plus y will be equal to 9. Now, if we add them together, we have x plus x is 2x is equal to because minus y plus y will still give us another zero. So, we have 4 plus 9 is 13. Now, if we solve this, you can still see from here that our x will be 13 divided by 2, which is not a natural number. Alright. This is not a natural number. We are going to still eliminate 4 times 9. Now, let's check the next one, which is 3 times 12. Okay. So, let's check 3 times 12. Now, if we do that, we are going to have x minus y multiplied by x plus y is equal to 3 times 12. So, from here, we have x minus y is equal to 3. x plus y is equal to 12. Now, if we add them together, we have x plus x is 2x. y plus plus y is 0. 3 plus 12 is 15. Now, you can still see very clearly that from here, our x is also a fraction. Therefore, this cannot serve as our x. And definitely, there is no need looking for y. Alright. Now, we are left with 2 multiplied by 18. Let's see if this will work. Okay. So, we have 2 multiplied by 18. That means we have x minus y is equal to 2 and x plus y is equal to 18. Now, if we add together, we are going to have x plus x is 2x and minus y plus plus y is 0 and 2 plus 18 is 20. So, we have 2x is equal to 20. Now, you can see clearly from here that if we divide both sides by 2, we will get our x to be equal to what? 10. And this is a natural number, right? It is a natural number. It is a counting number. Now, we need to find the corresponding value of y. To find the corresponding value of y, we can make use of either of these equations. So, let's take equation 1 that says that x take away y is equal to 2. Therefore, if we substitute the value of x, which is 10, into equation 1, we are going to have that 10 minus y is equal to 2. So, we have 10. If we transfer 2 to the other side, we are going to have 10 minus 2 is equal to and we take y to the other side. Minus y will cross to become plus y. Now, 10 minus 2 will give us 8 and this implies that y is equal to 8. So, in conclusion, our x is 10 and our y is equal to 8. But we need to verify this. We need to verify this to be sure that we are 100% correct. So, our x is 10 and our Now, let's see if that satisfies our equation. Now, x squared minus y squared. So, we have 10 squared minus 8 squared is equal to 36. We are verifying this now. Now, 10 squared is 100 and 8 squared is 64 is equal to 36. Now, 100 minus 64 is actually 36. So, now we have 36 is equal to 36 and that proves that the answer we got is correct. Thank you so much for watching and see you in my next video. Bye.