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Let a and b be relativly prime integers and let k be any intger. Show that b and a+bk are relatively prime Solution Suppose d = gcd(b,a + bk). Then d|b and d|(a + bk), and d divides any linear combination of b and a + bk. In particular, d divides (a + bk - kb) or d|a. Since d also divides b, it follows that d|gcd(a,b), or d|1. Therefore d = 1, so b and a + bk are relatively prime..

Let a and b be relativly prime integers and let k be any intger. Show that b and a+bk are
relatively prime
Solution
Suppose d = gcd(b,a + bk). Then d|b and d|(a + bk), and d divides any linear
combination of b and a + bk. In particular, d divides (a + bk - kb) or d|a. Since d also divides b, it
follows that d|gcd(a,b), or d|1. Therefore d = 1, so b and a + bk are relatively prime.

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