ZA-SAMO 2014 Problem 4

(a) Let $a,x,y$ be positive integers. Prove: if $x\ne y$ , the also $$ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).$$

(b) Show that there are no two positive integers $a$ and $b$ such that $$ab+\gcd(a,b)+\text{lcm}(a,b)=2014.$$