Periodic common refinement of floor-quotient partitions ($\ell =\mathrm{lcm}(a,b)$)

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 $\ell =\mathrm{lcm}(a,b)$. 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 $\{j\ell +c_i,\dots ,j\ell +c_{i+1}-1\}$ for $j\ge 0$ and $0\le i<r$. The pattern repeats, but blocks in different periods remain distinct equivalence classes.