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Q: What is 1000 b?
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What is the difference between the smallest perfect square larger than 1 million and the largest perfect square smaller than 1 million?

1 million = 1,000,000 = 1000² Which means we want: (1000 + 1)² - (1000 - 1)² , that is 1001² - 999² However, before we work that out, lets look at what we have. let a = 1000, b = 1, then: (1000 + 1)² - (1000 - 1)² = (a + b)² - (a - b)² Now (a + b)² = a² +2ab + b² and (a - b)² = a² - 2ab - b² → (a + b)² - (a - b)² = a² +2ab + b² - (a² - 2ab - b²) = 4ab As a = 1000, b = 1, this means (1000 + 1)² - (1000 -1)² = 4 × 1000 × 1 = 4000.


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There are infinitely many possible triplets.Suppose A is any positive real number that is less than 10 = cuberoot(1000).Let B be any positive real number between A and sqrt(1000/A).And let C = 1000/(A*B).This construction ensures that in each triplet (A, B, C), A < B < C so that each triplet is different and the same set is not being counted multiple times.Then, A*B*C = A*B*1000/(A*B) = 1000There are infintely many real numbers smaller than 10, there are therefore infinitely many choices for A.Then there are infinitely many real numbers between A and sqrt(1000/A), so there are infinitely many choices for B.Consequently, there are infinitely many possible solutions.


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