If (a,b)=(b,c)=(c,a)=1 prove the following...

[Guest]

If
(a,b)=(b,c)=(c,a)=1
prove that
(abc, ab+bc+ca)=1

[perfectmath]

Let [tex](a,b)=(b,c)=(c,a)=d[/tex], we will have $a=dm,b=dn,c=dk$ with [tex](m,n)=1;(n,k)=1;(k,m)=1[/tex].
Implies [tex]abc=d^3mnk[/tex] and [tex]ab+bc+ca=d^2(mn+nk+km)[/tex].
And [tex]d=1[/tex] then [tex](abc,ab+bc+ca)=(mnk,mn+nk+km)=1[/tex].

Is that solution right ??