Let $m$,$l$ be coprime integers where $m,l\geq 2$. For any integer $a$ and positive base $b \ (b\geq 2)$, let $ [a]_b $ denote the element of $\{0,\ldots, b-1\}$ that satisfies the equivalence $[a]_b \equiv a \bmod b$.

For any integer $n$, one can write $$ nl[l^{-1}]_m - nm[(-m)^{-1}]_l = n, $$ as Bézout's Identity yields $$ l[l^{-1}]_m - m[(-m)^{-1}]_l = 1 $$ (the existence of inverses is assured by the coprimality of $m$ and $l$).

**Question:** Under what conditions on $n$ does the equality
$$
l[nl^{-1}]_m - m[n(-m)^{-1}]_l = n
$$ hold?

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