Hello Codeforces,

Recently I was trying to make a number theory problem, I haven't framed the entire question yet but the core part goes something like this $$$-$$$ Given an integer $$$a$$$, find an integer $$$b$$$ ($$$ \le a $$$) such that $$$gcd(a,b)$$$ $$$+$$$ $$$lcm(a,b)$$$ is a prime number.

The condition $$$gcd(a,b)$$$ $$$+$$$ $$$lcm(a,b)$$$ is a prime number boils down to $$$gcd(a,b) = 1$$$ and $$$a \cdot b + 1$$$ is a prime where $$$b \le a$$$.

I tried brute-forcing $$$b$$$ by fixing $$$a$$$ and I found that there exists such $$$b$$$ for all $$$1 \le a \le 10^5$$$. However, I am not sure whether there exists such $$$b$$$ for all $$$a$$$. Does anyone have a formal or intuitive proof why this is happening and is there any approach to find $$$b$$$ given $$$a$$$ ?