If the area of a rectangle is divisible by $d$, must there exist a tiling of the region by polyominoes of size $d$?
It clearly isn't true that, if d divides the area of a rectangle, then there must exist a tiling of the rectangle using only a single polyomino of size d. (Counterexamples are easy to find; let the area equal five and d = 4, for instance.) But, given the same premise, it seems like it may be true that there exists a tiling of the rectangle using some combination of all of the various distinct d-ominoes. Indeed, intuitively, this feels like a paraphrase of what division means: take one geometrical representation of a quantity, and break it up into divisor quantities, of any possible geometrical configuration (i.e., any polyomino shape.)
Is this valid thinking, or is the rectangle constraint in the aforementioned statement too strong? I.e. -- it is definitely true that there exists some geometrical representation of the dividend that can be broken up into some combination of different geometrical representations of the divisor (by definition of division), but we can't be sure that a rectangle is the operative representation of the dividend. If this is the case, counterexamples would be appreciated.
Fix a positive integer $d$ and suppose you have an $a{\times}b$ rectangle $R$ where $a,b$ are positive integers such that $ab$ is a multiple of $d$.
Let $s=\gcd(a,d)$ and let $t={\large{\frac{d}{s}}}$.
Then we have $st=d$ and $t{\,\mid\,}b$, hence we can write $a=a_1s$ and $b=b_1t$ for some positive integers $a_1,b_1$.
It follows that $R$ can be tiled using $s{\times}t$ rectangular tiles.