Thinking about $ax + by = ab$ This is embarrassing because I know this is fairly simple, but I'm hitting a mental block and not having much luck with any references I'm aware of.
What properties does $ax + by = ab$ have? (a and b are integers)
I guess I'm sort of thinking...


*

*Is there always a solution? (I think so)

*How many solutions are there?

*Is there always a solution where neither x or y is zero? (not sure)


Does this sort of equation look familiar to anyone else? Does it remind you of anything? I feel like it must have some obvious properties.

$1.$ There is always a solution: we can write the greatest common denominator as $ax+by$, hence we just need to scale that solution in order to get a solution for $ab$. (If $a, b$ are zero it is trivial to see there is a solution)
$2.$ The number of solutions is infinite. Let $ax+by = \gcd(a, b)$. One of $a$ or $b$ must be non-zero. WLOG let $a$ be non-zero, then we can keep increasing $y$ by one and subtracting by $\frac{b}{\gcd(a, b)}(ax+by)$ to get another solution. (Again, the case where either $a, b$ are zero is trivial)
$3.$ The only time where we do not find a solution where both $x, y$ are zero is when only one of $a, b$ are zero.
