# Hindu-Arabic numerals and the newmore advanced algebra promoted by Fibonacci and Jordanusestern scholars, more inclined to theology and metaphysics,abor required to learn mathematics. We shall shortly see thatdanus were to enjoy a second life when revived by the Italianat has come to be called the Renaissance.Prove that if x +y is even, then the productxy(x +y)(x - y) is divisible by 24, and thatwithout this restriction, 4xy(x - y)(x + y) isdivisible by 24. [Hint: Consider that any integeris of the form 3k, 3k + 1, or 3k + 2 in showingthat 3 xy(x +y)(x - y). Similarly, because anyinteger is of the form 8k, 8k + 1, . . . , or 8k + 7,then 8xy(x - y)(x + y).](b)bigFind a square number such that when twice itsroot is added to it or subtracted from it, one(a)6.obtained other square numbers. In other words,solve a problem of the typex2-2x = 2in the rational numbers.(b)Find three square numbers such that the additionof the first and second, and also the addition ofall three squares, produces square numbers. Inother words, solve a problem of the type2x +y22 v2in the rational numbers. [Hint: Let x and y betwo relatively prime integers such that x2 yequals a square, say, x2 +y2 u2. Now note theidentity22 -12u212

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Asked Oct 29, 2019
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