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Proof: If gcd(a, b) = 1, then 1 = ax + by for some x, y; also cla + b mean some integer r. Thus, b = cr - a, a = cr - b, so 1 = a(x -y) + c(ry) = which implies that gcd (a, c) = gcd(b, c) = 1. %3D %3D %3D %3D Write the theorem statement here: Now, rewrite the proof below giving explanations for each step along the u were explaining this to someone. Justify each step.

Proof: If gcd(a, b) = 1, then 1 = ax + by for some x, y; also cla + b mean some integer r. Thus, b = cr - a, a = cr - b, so 1 = a(x -y) + c(ry) = which implies that gcd (a, c) = gcd(b, c) = 1. %3D %3D %3D %3D Write the theorem statement here: Now, rewrite the proof below giving explanations for each step along the u were explaining this to someone. Justify each step.

Advanced Engineering Mathematics
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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Proof: If gcd(a,b) = 1, then 1 = ax + by for some x, y; also cla + b means a + b = cr for some integer r. Thus, b = cr - a, a = cr – b, so 1 = a(x – y) +c(ry) = b(y – x) + c(rx), which implies that gcd(a, c) = gcd(b, c) = 1. %3D Write the theorem statement here: Now, rewrite the proof below giving explanations for each step along the way. Imagine you were explaining this to someone. Justify each step.
Transcribed Image Text:Proof: If gcd(a,b) = 1, then 1 = ax + by for some x, y; also cla + b means a + b = cr for some integer r. Thus, b = cr - a, a = cr – b, so 1 = a(x – y) +c(ry) = b(y – x) + c(rx), which implies that gcd(a, c) = gcd(b, c) = 1. %3D Write the theorem statement here: Now, rewrite the proof below giving explanations for each step along the way. Imagine you were explaining this to someone. Justify each step.
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