### Educational Content: Rational Numbers and Problem Solving---#### Problem 1: From Diophantus' ArithmeticaThe following problem appears in Book II of Diophantus' *Arithmetica*:**Problem Statement:**Divide a positive number, say 20, into a sum of two positive rational numbers $ x $ and $ y $ such that for some positive rational number $ z $, both $ z^2 + x $ and $ z^2 + y $ are squares (of rational numbers). For the sake of definiteness, write $ z^2 + x = (z+2)^2 $ and $ z^2 + y = (z+3)^2 $.---**Explanation:**This classic problem from Diophantus' work involves expressing a given number (in this case 20) as the sum of two rational numbers $ x $ and $ y $. Furthermore, when these numbers are added to the square of some rational number $ z $, the results must form perfect squares.**Given Equations:**1. $ z^2 + x = (z + 2)^2 $2. $ z^2 + y = (z + 3)^2 $**Instructions:**- Solve the given equations to find the rational numbers $ x $ and $ y $.- Verify that $ x + y = 20 $.This problem serves as an excellent exercise for students to practice:- The concepts of rational numbers.- Manipulation and simplification of algebraic expressions.- Problem-solving strategies in number theory.---Please use these instructions and the given values to work through the problem. Practice the algebraic manipulations to better understand how mathematicians like Diophantus approached such number theory problems.

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Description: ### Educational Content: Rational Numbers and Problem Solving---#### Problem 1: From Diophantus' ArithmeticaThe following problem appears in Book II of Diophantus' *Arithmetica*:**Problem Statement:**Divide a positive number, say 20, into a sum of t

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Divide a positive number, say 20, into a sum of two positive rational numbers x, y such that for some positive rational number z both z² + x and z2 + y are squares (of rational numbers). For the sake of definiteness, write z? + x = (z+ 2)² and z² + y = (z +3)².

Divide a positive number, say 20, into a sum of two positive rational numbers x, y such that for some positive rational number z both z² + x and z2 + y are squares (of rational numbers). For the sake of definiteness, write z? + x = (z+ 2)² and z² + y = (z +3)².

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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### Educational Content: Rational Numbers and Problem Solving

---

#### Problem 1: From Diophantus' Arithmetica

The following problem appears in Book II of Diophantus' *Arithmetica*:

**Problem Statement:**
Divide a positive number, say 20, into a sum of two positive rational numbers $ x $ and $ y $ such that for some positive rational number $ z $, both $ z^2 + x $ and $ z^2 + y $ are squares (of rational numbers). For the sake of definiteness, write $ z^2 + x = (z+2)^2 $ and $ z^2 + y = (z+3)^2 $.

---

**Explanation:**

This classic problem from Diophantus' work involves expressing a given number (in this case 20) as the sum of two rational numbers $ x $ and $ y $. Furthermore, when these numbers are added to the square of some rational number $ z $, the results must form perfect squares.

**Given Equations:**

1. $ z^2 + x = (z + 2)^2 $
2. $ z^2 + y = (z + 3)^2 $

**Instructions:**

- Solve the given equations to find the rational numbers $ x $ and $ y $.
- Verify that $ x + y = 20 $.

This problem serves as an excellent exercise for students to practice:
- The concepts of rational numbers.
- Manipulation and simplification of algebraic expressions.
- Problem-solving strategies in number theory.

---

Please use these instructions and the given values to work through the problem. Practice the algebraic manipulations to better understand how mathematicians like Diophantus approached such number theory problems.

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