20. Confirm the following properties of the greatest common divisor:(a) If gcd(a , b) = 1, and gcd(a , c) = 1, then gcd(a , bc) = 1.[Hint: Because 1 = ax + by = au + cv for some x, y, u, v,1 = (ax + by)(au + cv) = a(aux + cvx +byu)+bc(yv).](b) If gcd(a , b) = 1, and c | a, then gcd(b , c) = 1.%3D

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Let (Z8, +8) and H={0,2,4,6} .Find the normalizer of (H, +8) in (Z8, +8).
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20. Confirm the following properties of the greatest common divisor: (a) If gcd(a , b) = 1, and gcd(a , c) = 1, then gcd(a , bc) = 1. [Hint: Because 1 = ax + by = au + cv for some x, y, u, v, 1 = (ax + by)(au + cv) = a(aux + cvx + byu)+bc(yv).] (b) If gcd(a , b) = 1, and c|a, then gcd(b , c) = 1.