Prove that gcf(p,q) = gcf(p+q,lcm(p,q))?

Question

Prove that gcf(p,q) = gcf(p+q,lcm(p,q))?

Answer 1

Let  p = (b₁b₂*…*b_n)S

and q = (b₁b₂*…*b_n)R

 

where  b₁b₂*…*b_n = common prime factors of p and q

   and  S = all other factors of p not found in q

          R = all other factors of q not found in p   (so  R∩S = {ø} )

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By definition, gcf(p,q) =  b₁b₂*…*b_n

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p+q = (b₁b₂*…*b_n)S + (b₁b₂*…*b_n)R

 

lcm(p,q) = pq/gcf(p,q) = ((b₁b₂*…*b_n)²) *R*S/(b₁b₂*…*b_n) = (b₁b₂*…*b_n)*R*S

 

gcf(p+q, lcm(p,q)) = gcf( (b₁b₂*…*b_n)(S+R),  (b₁b₂*…*b_n)*R*S) = b₁b₂*…*b_n  (This completes the proof)

 

In the last expression, keep in mind neither  R+S nor R*S have any common factors to both p and q.