# 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.