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.

You may want to try and prove first that
$\operatorname{gcd}(a, b) = \operatorname{gcd}(b, a - b)$
This identity should be enough to get you rolling. You will just have to be able to use it correctly.
Alternatively, try this approach.
Suppose $d $ is the gcd of $a $ and $b $. Show it divides $a+b $ and $\operatorname {lcm}(a, b) $.
Then suppose $d'$ is the gcd of $a+b $ and $\operatorname {lcm}(a, b) $. Show  $d'$ divides both $a $ and $b $. Hence $d $ divides $d'$ which divides $d $, meaning they must be equal.
