Please provide clear, complete, and neat solutions step by step for the given problem Thank you Assume c gcd(a, b), c ax 0 by 0 , x 0 , y 0 Z, a cr and b cs, r, s Z Prove c a(x 0 st) b(y 0 rt) for all t Z (Hint Show that c(ast brt) 0 and conclude that ast brt 0 )

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Question: Please provide clear, complete, and neat solutions step by step for the given problem. Thank you: Assume c = gcd(a, b), c = ax 0

Please provide clear, complete, and neat solutions step by step for the given problem. Thank you:

Assume c = gcd(a, b), c = ax₀+by₀, x₀, y₀ Z, a = cr and b = cs, r, s Z. Prove: c = a(x₀ + st) + b(y₀ rt) for all t Z. (Hint: Show that c(ast brt) = 0 and conclude that ast brt = 0.)

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