Question 86
(a) Let $a$, $x$, $y$ be positive integers. Prove: if $x \neq y$, then also
$$
ax + \gcd(a, x) + \text{lcm}(a, x) \neq ay + \gcd(a, y) + \text{lcm}(a, y).
$$
(b) Show that there are no two positive integers $a$ and $b$ such that
$$
ab + \gcd(a, b) + \text{lcm}(a, b) = 2014.
$$
$$
ax + \gcd(a, x) + \text{lcm}(a, x) \neq ay + \gcd(a, y) + \text{lcm}(a, y).
$$
(b) Show that there are no two positive integers $a$ and $b$ such that
$$
ab + \gcd(a, b) + \text{lcm}(a, b) = 2014.
$$