(fraction in lowest terms)

Hi my names jacob and im having some problems with these questions, just wondering if anyone can help???

.Letr 2Q.

(a) Prove that

rhas a least possible denominatorn, i.e.,r=m=nfor somem 2Z

and

n 2N and that for non′ < nis it possible to writer=m′=n′. We say that

such

nisdenominator of r in lowest termsand the fractionm=nis thelowest terms

representation of

ras a fraction.

Hint. Introduce the set

S:=

fl 2N :9 k 2Z :r=k=lg :(3.3.1)

Show that

Sis non-empty and apply the well ordering property of N.

(b) Show that if

m=nis the lowest term fraction ofr ̸= 0 thenmandnare co-prime.

(c) Show that there is a unique pair (

m; n)2Z N for whichr=m=nin lowest

terms.