Answer:
One proof can be as follows:
Step-by-step explanation:
We have that [tex]g.c.d(a,b)=1[/tex] and [tex]a=cp, b=dq[/tex] for some integers [tex]p, q[/tex], since [tex]c[/tex] divides [tex]a[/tex] and [tex]d[/tex] divides [tex]b[/tex]. By the Bezout identity two numbers [tex]a,b[/tex] are relatively primes if and only if there exists integers [tex]x,y[/tex] such that
[tex]ax+by=1[/tex]
Then, we can write
[tex]1=ax+by=(cp)x+(dq)y=c(px)+d(qy)=cx'+dy'[/tex]
Then [tex]c[/tex] and [tex]d[/tex] are relatively primes, that is to say,
[tex]g.c.d(c,d)=1[/tex]
