Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.2

 2. Show that if a and b are positive integers, then there is a smallest positive integer of the form a – bk, k \in Z.

Solution:

Let a and b be positive integers and let S = { n : n is a positive integer and n = a – bk for some k \in \mathbb{Z}. Now S is non empty since a + b = a – b( – 1) is in S . By the Well Ordering Principle S has a least element.

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