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Consider two sets of numbers.

$E_1=\{1,1,-1,...\}$

$E_2=\{1,-1,-1,...\}$

Let the number of elements in $E_1$ be $n_1$ and the number of elements in $E_2$ be $n_2$. The number of 1's and -1's in each set may vary for different cases. The maximum of total number of elements for both sets, i.e. $n_1 + n_2$ that could be present is $N$.

$max(n_1+n_2)=N$

Let $E_1(i)$ denote the $i^{th}$ element of $E_1$, $i=1, ..., n_1$, $E_2(j)$ denote the $j^{th}$ element of $E_2$, $j=1, ..., n_1$, $\sum_{i=1}^{n_1} E_1(i) = a$ and $\sum_{j=1}^{n_2} E_2(j) = b$.

$a$ and $b$ should satisfy the relationships,

$a \equiv 0 (mod p)$

$b \equiv 0 (mod q)$, where $p,q$ are two distinct primes. These two relationships are satisfied when $n_1+n_2 <N$ as well as when $n_1+n_2 =N$.

Suppose I do not know,

  1. the elements of the two sets $E_1$ and $E_2$
  2. $n_1$ and $n_2$

But I know $p, q$ and $N$ (Actually $N=pq$). Then, is there a method to determine the values of $a$ and $b$ using the above two equations for the case $n_1+n_2=N$ (i.e. the maximum case)?

It is not clear to me, this is not like considering two normal simultaneous equations but here congruence relations are present.

Thanks a lot in advance.

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