Math

[Guest]

Prove that if a,b,c,d are positive integers,
(b+c) is a factor of (bd) then
gcd(b+c,cd+ab+ac+1)=1.

[shyamjayakannan]

[tex]bd=(b+c)d-cd[/tex] and since [tex]b+c[/tex] is a factor of [tex]bd[/tex], [tex]b+c[/tex] must also be a factor of [tex]cd[/tex].
So, [tex]cd+ab+ac+1=cd+a(b+c)+1=k(b+c)+1[/tex], where k is some integer.
So, [tex]\boxed{gcd(b+c,cd+ab+ac+1)=1}[/tex]