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Hindu-Arabic numerals and the new more advanced algebra promoted by Fibonacci and Jordanus estern scholars, more inclined to theology and metaphysics, abor required to learn mathematics. We shall shortly see that danus were to enjoy a second life when revived by the Italian at has come to be called the Renaissance. Prove that if x +y is even, then the product xy(x +y)(x - y) is divisible by 24, and that without this restriction, 4xy(x - y)(x + y) is divisible by 24. [Hint: Consider that any integer is of the form 3k, 3k + 1, or 3k + 2 in showing that 3 xy(x +y)(x - y). Similarly, because any integer is of the form 8k, 8k + 1, . . . , or 8k + 7, then 8xy(x - y)(x + y).] (b) big Find a square number such that when twice its root is added to it or subtracted from it, one (a) 6. obtained other square numbers. In other words, solve a problem of the type x2-2x = 2 in the rational numbers. (b) Find three square numbers such that the addition of the first and second, and also the addition of all three squares, produces square numbers. In other words, solve a problem of the type 2 x +y22 v2 in the rational numbers. [Hint: Let x and y be two relatively prime integers such that x2 y equals a square, say, x2 +y2 u2. Now note the identity 2 2 -1 2 u21 2

Hindu-Arabic numerals and the new more advanced algebra promoted by Fibonacci and Jordanus estern scholars, more inclined to theology and metaphysics, abor required to learn mathematics. We shall shortly see that danus were to enjoy a second life when revived by the Italian at has come to be called the Renaissance. Prove that if x +y is even, then the product xy(x +y)(x - y) is divisible by 24, and that without this restriction, 4xy(x - y)(x + y) is divisible by 24. [Hint: Consider that any integer is of the form 3k, 3k + 1, or 3k + 2 in showing that 3 xy(x +y)(x - y). Similarly, because any integer is of the form 8k, 8k + 1, . . . , or 8k + 7, then 8xy(x - y)(x + y).] (b) big Find a square number such that when twice its root is added to it or subtracted from it, one (a) 6. obtained other square numbers. In other words, solve a problem of the type x2-2x = 2 in the rational numbers. (b) Find three square numbers such that the addition of the first and second, and also the addition of all three squares, produces square numbers. In other words, solve a problem of the type 2 x +y22 v2 in the rational numbers. [Hint: Let x and y be two relatively prime integers such that x2 y equals a square, say, x2 +y2 u2. Now note the identity 2 2 -1 2 u21 2

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 64RE
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Hindu-Arabic numerals and the new more advanced algebra promoted by Fibonacci and Jordanus estern scholars, more inclined to theology and metaphysics, abor required to learn mathematics. We shall shortly see that danus were to enjoy a second life when revived by the Italian at has come to be called the Renaissance. Prove that if x +y is even, then the product xy(x +y)(x - y) is divisible by 24, and that without this restriction, 4xy(x - y)(x + y) is divisible by 24. [Hint: Consider that any integer is of the form 3k, 3k + 1, or 3k + 2 in showing that 3 xy(x +y)(x - y). Similarly, because any integer is of the form 8k, 8k + 1, . . . , or 8k + 7, then 8xy(x - y)(x + y).] (b) big Find a square number such that when twice its root is added to it or subtracted from it, one (a) 6. obtained other square numbers. In other words, solve a problem of the type x2-2x = 2 in the rational numbers. (b) Find three square numbers such that the addition of the first and second, and also the addition of all three squares, produces square numbers. In other words, solve a problem of the type 2 x +y22 v2 in the rational numbers. [Hint: Let x and y be two relatively prime integers such that x2 y equals a square, say, x2 +y2 u2. Now note the identity 2 2 -1 2 u21 2
Transcribed Image Text:Hindu-Arabic numerals and the new more advanced algebra promoted by Fibonacci and Jordanus estern scholars, more inclined to theology and metaphysics, abor required to learn mathematics. We shall shortly see that danus were to enjoy a second life when revived by the Italian at has come to be called the Renaissance. Prove that if x +y is even, then the product xy(x +y)(x - y) is divisible by 24, and that without this restriction, 4xy(x - y)(x + y) is divisible by 24. [Hint: Consider that any integer is of the form 3k, 3k + 1, or 3k + 2 in showing that 3 xy(x +y)(x - y). Similarly, because any integer is of the form 8k, 8k + 1, . . . , or 8k + 7, then 8xy(x - y)(x + y).] (b) big Find a square number such that when twice its root is added to it or subtracted from it, one (a) 6. obtained other square numbers. In other words, solve a problem of the type x2-2x = 2 in the rational numbers. (b) Find three square numbers such that the addition of the first and second, and also the addition of all three squares, produces square numbers. In other words, solve a problem of the type 2 x +y22 v2 in the rational numbers. [Hint: Let x and y be two relatively prime integers such that x2 y equals a square, say, x2 +y2 u2. Now note the identity 2 2 -1 2 u21 2
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