C 288 8. The algebraic identity (a2+b2)(c2 +d2) = (ac + bd)2 + (ad - bc)2 (ad+ bc(ac - bd)2 12. 11 appears in the Liber Quadratorum. Establish this identity and use it to express the integer 481 =...

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Extracted text: C 288 8. The algebraic identity (a2+b2)(c2 +d2) = (ac + bd)2 + (ad - bc)2 (ad+ bc(ac - bd)2 12. 11 appears in the Liber Quadratorum. Establish this identity and use it to express the integer 481 = 13.37 as the sum of two squares in two different ways. 13. Given rational numbers a and b, find two other rational numbers x and y such that a2b2 =x2 + y2. [Hint: Choose any two integers c and d for which c2+ d2 is a square; now write (a2 + b2)(c d2) as a sum of two 9. (a) squares.] (b) Illustrate part (a) by expressing 61 = 52 + 62 as the sum of squares of two rational numbers. t 10. Solve the following problem, which is one of the tournament problems that John of Palermo posed to Fibonacci. Each of three men owned a share in a pile of money, their shares being Each man took some money at random until nothing was left. The first man afterward returned of what he had taken,the second amount thus returned was divided into three equal parts and given to each man, each one had what he was originally entitled to. How much money was there in the pile at the start, and how much did each man take? [Hint: Let t denote the original sum, u the amount each man received when the money left in the pile was divided equally, and x, y, and z the amount of the total. and 3 . 2 , and the third. When the Tо fo (а) took. Then