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What are necessary and sufficient conditions for this System to have a solution?

$x \equiv 2a_1^2(mod \ 2a_1-1)$

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$x \equiv 2a_m^2(mod \ 2a_m-1)$

I know that for each $i, j$, $a_i^2 \equiv a_j^2 (mod \ gcd(2a_i-1, 2a_j-1))$

So how can i start from here and find necessary and sufficient condition for this system?

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By this answer the system is solvable iff $\,\gcd(2a_i-1,2a_j-1)\mid 2(a_i^2-a_j^2)\,$ for $\,i\neq j.\,$ But the gcd is odd so coprime to $2$ so it's equivalent to $\,\gcd(2a_i-1,2a_j-1)\mid a_i^2-a_j^2,\,$ i.e. your hypothesis.

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