Hi my names jacob and im having some problems with these questions, just wondering if anyone can help???
(fraction in lowest terms). Let r 2 Q.
(a) Prove that
r has a least possible denominator n, i.e., r = m=n for some m 2 Z
n 2 N and that for no n′ < n is it possible to write r = m′=n′. We say that
n is denominator of r in lowest terms and the fraction m=n is the lowest terms
r as a fraction.
. Introduce the set
:= fl 2 N : 9 k 2 Z : r = k=lg : (3.3.1)
S is non-empty and apply the well ordering property of N.
(b) Show that if
m=n is the lowest term fraction of r ̸= 0 then m and n are co-prime.
(c) Show that there is a unique pair (
m; n) 2 Z N for which r = m=n in lowest