Subject: The function ax+gcd(a,x)+lcm(a,x) djb86 wrote: (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 \[ab+\gcd(a,b)+\text{lcm}(a,b)=2014.\] (a) if we have $ax+\gcd(a,x)+\text{lcm}(a,x)= ay+\gcd(a,y)+\text{lcm}(a,y)$ we will have $\gcd(a,x)-gcd(a,y)=a(y-x)+\text{lcm}(a,y)-\text{lcm}(a,y)$ that can be divided by $a$ but $a$ doesn't divide $\gcd(a,x)-gcd(a,y)$ that is between $-a+1$ and $a+1$ (b) $ab+\gcd(a,b)+\text{lcm}(a,b)=2ab+1$ is odd and $2014$ is even