Question 12. Use the Euclidean algorithm to represent the greatest common diviso

Question

Question 12. Use the Euclidean algorithm to represent the greatest common divisor of two numbers a and b as a linear combination az + by (a, b, x, y are integers). As an example you may want to work out, take a = 826, b 732. Question 13. Let p be a prime and let a and b be integers such that p is a divisor of ab. Prove that then p is a divisor of a or b ESSAY QUESTIONS Question 14. Write a brief essay about the beginnings of mathematical proof in Greece. Question 15. Explain why the existence of incommensurable distances causes trouble for the Pythagore Question 16. Explain how Eudoxus resolved the crisis created by the existence of incommensurable Question 17. Briefly describe the three classical problems of Athens (doubling of the cube, squaring the ans (Hint: it invalidated some of their theories, such as the theory of similar triangles). distances circle, trisecting the angle)

Explanation / Answer

12. Given Numbers a = 826 and b = 732.

then Linear combination using Euclidean algorithm:

826 = 732*1 + 94
732 = 94*7 + 74
94 = 74*1 + 20
74 = 20*3 + 14
20 = 14*1 + 6
14 = 6*2 + 2
6 = 2*3 + 0

Linear Combination: 2 = 123*732 + -109*826

and GCD = 2

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