Let $\lceil x\rceil$ denote the value of the ceiling function at $x.$ Consider two distinct odd primes $p$ and $q.$ I want to find integers $n$ (if exixts) between $0$ and $pq$, and non-negative integers $a,b, c, d$ such that it satisfies the following equations together:
$$a+2c+ \lceil \frac{n}{p}\rceil = \lceil \frac{n+a+2b+2c+2d}{p}\rceil$$
$$a+2d +\lceil \frac{n}{q}\rceil = \lceil \frac{n+a+2b+2c+2d}{q}\rceil$$
$$a+ \lceil \frac{n}{pq}\rceil = \lceil \frac{n+a+2b+2c+2d}{pq}\rceil$$
Thank you in advance. Any help will be appreciated.