Basic terminology question from a non-mathematician. I started trying to express this with mathematical terms, but decided any potential errors might be more frustrating than the imprecision of written language.
Let's say I have a rectangle $Q$ with sides with integer lengths, $a$ and $b$. The rectangle $Q$ can be scaled, dividing the length of each side by the same rational scaling factor, resulting in a second rectangle $Q_n$ with the same aspect ratio and with sides of integer length. Is there a name for the relationship between these two rectangles, like "$Q_n$ is an integer factor of $Q$?"
[In case you haven't guessed it, I'm doing some image processing and want to talk about ways of minimizing distortion.]
Both your similar rectangles are integer multiples (scalings) of some common integer rectangle which has sides of coprime lengths (which might possibly also be identical one of the two rectangles, of course).
So there is a "primitive" rectangle $Q_p$ that both $Q$ and $Q_n$ are multiples of. Taking $g_Q= \gcd(a, b)$, $Q_p$ has sides $\left(\frac{a}{g_Q}, \frac{b}{g_Q} \right)$.
To extend the language of integers into this concept, it's not (necessarily) that $Q_n$ is a factor-rectangle of $Q$, or vice versa, but that they have a common factor-rectangle ($Q_p$).