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I have been reading up on finding incongruent solutions of quadratic congruences and have stumbled upon an answer to a question asked here. The answer I am confused about is the following:

"if you have $x^2 \equiv 23 \pmod {77}$, then we need to look at $x^2 \equiv 23 \pmod 7$ and $x^2 \equiv 23 \pmod{11}$ i.e. $x^2 \equiv 2 \pmod 7$ and $x^2 \equiv 1 \pmod{11}$. $$x^2 \equiv 2 \pmod7 \implies x \equiv \pm 3 \pmod 7$$ Similarly, $$x^2 \equiv 1 \pmod{11} \implies x \equiv \pm 1 \pmod{11}$$

Hence, $$x \equiv 3 \pmod 7 \text{ and } x \equiv 1 \pmod{11} \implies x \equiv 45 \pmod{77}$$ $$x \equiv -3 \pmod 7 \text{ and } x \equiv 1 \pmod{11} \implies x \equiv 67 \pmod{77}$$ $$x \equiv 3 \pmod 7 \text{ and } x \equiv -1 \pmod{11} \implies x \equiv 10 \pmod{77}$$ $$x \equiv -3 \pmod 7 \text{ and } x \equiv -1 \pmod{11} \implies x \equiv 32 \pmod{77}$$

Hence, $$x \equiv \pm 10, \pm 32 \pmod{77}"$$ I understand most of the answer up until the point where the chosen answers are only $$ \pm 10, \pm 32 \pmod{77}.$$

Can someone help me understand why 45 and 67 are not proper answers? I attempted to understand the differences between each answer but as far as I see, they have the same properties relative to the original question. I checked the gcd for each answer and all are relatively prime. (P.S:I would leave a comment on the actual post but it was active back in 2013 and I am unsure if I would get a response leaving a comment on a post that old)

Michael Hardy
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Pauliene
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    $\bmod 77!:,\ 45\equiv -32,,\ 67\equiv -10,,$ so they are congruent to the listed solutions. – Bill Dubuque Dec 07 '21 at 02:39
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    See here and here for further detail on the underlying method (lifting roots using CRT). – Bill Dubuque Dec 07 '21 at 02:54
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    Thank you so much for your responses! So then it would also be correct to have the final answers stay as 45,67,10 and 32 right? – Pauliene Dec 07 '21 at 03:16
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    Yes, you can choose any integer congruent to a root as its representative. The OP's choice corresponds to used a balanced (least-magnitude) system of reps, e.g. $: \pm{0,1,2,3}\pmod 7\ \ $ – Bill Dubuque Dec 07 '21 at 08:17

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$45$ and $67$ are perfectly proper answers, and in some contexts (but not all contexts) its conventional usage would require answers to be in the set $\{0,1,2,\ldots,77-1\},$ so that that way of writing the answers would be required. However, it seems that someone wanted to emphasize the role of $\text{“} {\pm} \text{”}$ in the solutions to $x^2\equiv\cdots,$ and that's certainly a thing one should be aware of.

Michael Hardy
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