Prove $\gcd(a,b)=\gcd(a+b,\operatorname{lcm}(a,b))$ Any ideas on how to prove this equality?
I tried various methods, using properties of gcd and lcd, but I can't prove it.

Let gcd(a,b)=d. a=ds , b=de . a+b=d(e+s). lcm(a,b)=ab/d=des. Assume x greater than d divides des,d(e+s). By the first it divides one of e and s. But if it divides one, it must divide both by the second. But hten dx is the gcd a contradiction.
