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Can You Solve for (a + b)? Math Olympiad Question
In this video, we will solve a math Olympiad question that goes as follows: a + ab + b = 2020 . We will use some algebraic manipulations to find the solutions of this equation. Firstly, we can factorize this equation by using the distributive property of multiplication over addition. So we can rewrite the equation as a(1+b) + b = 2020. Next, we can use the fact that the sum of two numbers is equal to the product of their greatest common divisor and their least common multiple. In other words, a + b = gcd(a,b) x lcm(a,b). Therefore, we can rewrite our equation as a(1+b) + gcd(a,b) x lcm(a,b) = 2020. Now, we need to consider the prime factorization of 2020, which is 2^2 x 5 x 101. We can use this to find the possible values of gcd(a,b) and lcm(a,b). We can see that gcd(a,b) can only be 1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010, 2020. Similarly, lcm(a,b) can only be 1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010, 2020. Finally, we can substitute the possible values of gcd(a,b) and lcm(a,b) in our equation and solve for a and b. We can use trial and error or a systematic approach to find all the solutions. In this video, we will solve this problem step-by-step, explaining the logic and reasoning behind each step. So, if you re ready to dive into some algebraic manipulations and prime factorizations, let s get started! :::: Visit my YouTube channel @ Math with Mershscore ( [ 🔗 Click ‼️ ] )
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