= Periodic common refinement of floor-quotient partitions
{title2=$\ell=\operatorname{lcm}(a,b)$}
For positive <integers> $a,b$, partition the <nonnegative integers> by equality of $\lfloor n/a\rfloor$ and separately by equality of $\lfloor n/b\rfloor$. The <intersection of equivalence relations> gives nonempty intersections of these consecutive blocks. Its boundaries occur at multiples of either $a$ or $b$.
Let $\ell=\operatorname{lcm}(a,b)$. Translating by $\ell$ increases the two floor quotients by the fixed <integers> $\ell/a$ and $\ell/b$, so it preserves all within-block equalities and all block boundaries. Thus the refinement repeats in intervals of length $\ell$. If $0=c_0<c_1<\cdots<c_r=\ell$ lists the multiples of $a$ or $b$ within one period, its blocks are exactly $\{j\ell+c_i,\ldots,j\ell+c_{i+1}-1\}$ for $j\ge0$ and $0\le i<r$. The pattern repeats, but blocks in different periods remain distinct <equivalence classes>.
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