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,
- the elements of the two sets $E_1$ and $E_2$
- $n_1$ 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.