Suppose $p,p',p'',q,q',q'',q_1,q_1',q_1'',r$ are integers with $0<q,q',q_1,q_1',q_1''<r<p,p',p''$ and $$pq+p'q'+p''q''=pq_1+p'q_1'+p''q_1''=t$$
If $m=p\bmod r$, $m'=p'\bmod r$, $m''=p''\bmod r$ where $0\leq m,m',m''<r$ holds then there are $k,k',k''$ such that $$p=m+kr,\quad p'=m'+k'r,\quad p''=m''+k''r$$
Is there always two different integers $n,n_1$ such that $$mq+m'q'+m''q''=\ell+n r$$ $$mq_1+m'q_1'+m''q_1''=\ell+n_1 r$$ holds where $\ell=t\bmod r$ and $0\leq \ell<r$?
Can we give explicit formula for $n,n_1$?