Harvard University Entrance Exam Question | Can you solve ?

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[00:00:00]Hello, welcome back once again.

[00:00:03]Today, we have this interesting math problem from Harvard University.

[00:00:07]We're given that A, B, and C are integers greater than zero.

[00:00:12]And we have these two equations: A + BC = 2024, AB + C = 2023.

[00:00:20]So, let's get started.

[00:00:22]Now, let's subtract two equations, these two equations.

[00:00:27]So, let's start from here. So, we're going to have here A + BC minus from the left-hand side AB + C.

[00:00:38]So, this is equal to from the right-hand side 2024 minus 2023.

[00:00:45]Awesome.

[00:00:47]Now, here let's regroup this. So, this is going to be A minus AB, since we know that A is attached here.

[00:00:56]Then here plus BC minus C, since we know that C is attached here. So, this is equal to one.

[00:01:05]From here, you can factor out the common, which is A. So, it will be A into bracket 1 minus B.

[00:01:12]From here, if you factor out C, you're going to have B minus one. But we need this second bracket to look exactly to exactly as what is here, right? So, it will look exactly with this. So, we're going to factor out negative C. So, if you factor out negative C, we get negative B + 1.

[00:01:32]So, this is equal to one.

[00:01:34]So, here we get A into bracket 1 minus B.

[00:01:39]Then minus C into bracket 1 minus B.

[00:01:43]This is equal to one.

[00:01:45]So, from here this is factorized as A minus C multiplied by 1 minus B is equal to now one. We can factor one as 1 * 1 and again -1 * -1. So, these are the only two possibilities. Remember that a, b, and c are integers.

[00:02:05]So, we know that we know that 1 can be written in the form 3/4 * 4/3. But, we don't need this. These are rationals, right? But, we need integers. Therefore, there are two possibilities: 1 * 1 and -1 * -1.

[00:02:22]Now, when a - c * 1 - b is equal to 1 * 1, we get a - c is equal to 1 and 1 - b is equal to 1. From this second equation, you can see that b is equal to 0. So, we don't need this.

[00:02:40]Remember that a, b, and c are greater than 0.

[00:02:43]So, we use the second one here. So, here we have a - c is equal to -1. We have 1 - b is equal to -1. You can see now from this second equation, b is equal to 1 - -1, giving us 2.

[00:02:59]Now, from this equation, add c to both sides, we get a is equal to c - 1.

[00:03:05]Now, we have b is equal to 2 and a in terms of c is equal to c - 1. So, from the first equation, we can substitute the value of a and b to solve for c. So, remember we have a + bc is equal to 2024.

[00:03:22]So, a's value, that is c - 1. So, we get c - 1 + bc will be 2c.

[00:03:29]This is equal to 2024.

[00:03:32]So, from here we get c + 2c, that is 3c - 1 is equal to 2024.

[00:03:41]So, when you add 1 to both sides, we get 3c is equal to 2024 + 1, giving us 2025.

[00:03:48]So, divide both sides by 3 to get here C is equal to 675.

[00:03:58]Then what is our A value? Remember A here is going to be equal to C minus 1, which is 675 minus 1, giving us 674.

[00:04:09]So therefore we have the following solutions for A, B, and C. So we have A, B, and C integers greater than zero. It is going to be A that is 674, B 6 Oh my god, B is 2, right? So B is 2, and C 675.

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