Intuition - Linear Congruence Theorem 
Let a and b be integers (not both 0) with greatest common divisor d.
  Then an
  integer $c = ax + by$ for some $x, y \in Z$  $\iff d|c$.
In particular, d is the least positive integer of the form ax +by.

Is there intuition? Or illustration? I keep forgetting which variable is supposed to go. I'm not querying proofs. 
Withal, if I write $ax + by = c$ as $ax \equiv c \; (mod \, b)$, then an error is even more likely! I can't remember if it's  $c|d$, $b|c$, $c|a$, or some other wrong combination...
Origin - Elementary Number Theory, Jones, p10, Theorem 1.8

You have two integers $a$ and $b$ (not both $0$) with the greatest common divisor $d$. It means that $d|a$ and $d|b$. As we know, if $d|a$ and $d|b$ then $d|ax+by$ for all $x,y\in\mathbb{Z}$. Furthermore, one wonders what the set $$S=\{ax+by:x,y\in\mathbb{Z}\}.$$
Of course, $d$ divides each element in $S$ since is one of the common factor(s) between $a$ and $b$ which is equivalent to the necessary part of your statement.
The sufficient part of your statement is motivated by Bezout's_identity.
