Chapter 4.2, Problem 36E

1. Let g be the greatest common divisor of integers m and n, not both 0. Show that there are infinitely many pairs a, b of integers such that $g=am+bn$.

2. Suppose m and n are relatively prime integers each greater than 1.

   1. Show that there exist unique integers a, b with $0<b<m$ such that $am+bn=1$.

   2. Show that there exist unique integers a, b with $0<a<n$ and $0<b<m$ such that $am=bn+1$.