HW2sol - Homework 2: Math 3360 Applicable Algebra Chapter...

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Homework 2: Math 3360 Applicable Algebra Chapter 3: Euclid’s Algorithm 35ii. Using Euclid’s Algorithm, find the greatest common divisor of 21063 and 43137. 43137 = 2 · 21063 + 1011 21063 = 20 · 1011 + 843 1011 = 1 · 843 + 168 843 = 5 · 168 + 3 168 = 56 · 3 Thus gcd (43137 , 21063) = 3. 39ii. Find d the greatest common divisor, and find r,s so that ar + bs = d where a and b are 242 and 1870. Solution: Using the Euclidean algorithm we get: 1870 = 7 · 242 + 176 242 = 1 · 176 + 66 176 = 2 · 66 + 44 66 = 1 · 44 + 22 44 = 2 · 22 Thus gcd (1870 , 242) = 22. Doing the Euclidean algorithm backwards we get: 22 = 66 - 1 · 44 = 66 - 1 · (176 - 2 · 66) = 3 · 66 - 1 · 176 = 3 · (242 - 1 · 176) - 1 · 176 = 3 · 242 - 4 · 176 = 3 · 242 - 4 · (1870 - 7 · 242) = 31 · 242 - 4 · 1870 Thus we have r = 31 and s = - 4. 55. Claim : For a,b natural numbers, consider the set J of all positive integers of the form ar + bs for r,s . Since J is a nonempty set of the natural numbers, by well ordering
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HW2sol - Homework 2: Math 3360 Applicable Algebra Chapter...

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