I was doing some number theory problems and while working on a problem I came up with an useful generalization.
I know it is not very special but it's worth sharing in my view.

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Nice result! However, I think you can greatly improve your writing and formatting; it seems very hard to read and follow, but I would probably chalk that down to your possible lack of experience with mathematical writing.

I know but I cannot figure out how to use Latex .(I have absolutely 0 coding experience as I had never taken up computer science.)
I try to use it but the writing gets all messed up. Maybe someone will type it out on Latex.

I'm not just talking about LaTeX (I can understand that you may not know how to use it); I found the wording of the proof quite hard to follow. For example, instead of saying lines 2 to 7, you could say the following:

"As $2$ and $a$ are coprime, we would like to find $x \equiv a^n$ (here, the "$\equiv$" sign means "defined to be") such that

$x = 1 \quad (\text{mod } 2)$

and

$x = 0 \quad (\text{mod } a).$

We note that this solution is unique by the Chinese Remainder Theorem."

If you were using LaTeX, you'd also want to take advantage of the Theorem/Proof layout, but even if you don't you would need to separate the theorem from the result. The subsequent six lines could also be rewritten in a similar flavour.

Somewhat more directly, it just boils down to $a^2 \equiv a \pmod{2a}$. This follows since $a (a-1) \equiv 0 \pmod{2a}$ as the first term is a multiple of $a$ and the second term is a multiple of 2.

Easy Math Editor

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TopNewestNice result! However, I think you can greatly improve your writing and formatting; it seems very hard to read and follow, but I would probably chalk that down to your possible lack of experience with mathematical writing.

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I know but I cannot figure out how to use Latex .(I have absolutely 0 coding experience as I had never taken up computer science.) I try to use it but the writing gets all messed up. Maybe someone will type it out on Latex.

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I'm not just talking about LaTeX (I can understand that you may not know how to use it); I found the wording of the proof quite hard to follow. For example, instead of saying lines 2 to 7, you could say the following:

"As $2$ and $a$ are coprime, we would like to find $x \equiv a^n$ (here, the "$\equiv$" sign means "defined to be") such that

$x = 1 \quad (\text{mod } 2)$

and

$x = 0 \quad (\text{mod } a).$

We note that this solution is unique by the Chinese Remainder Theorem."

If you were using LaTeX, you'd also want to take advantage of the Theorem/Proof layout, but even if you don't you would need to separate the theorem from the result. The subsequent six lines could also be rewritten in a similar flavour.

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Somewhat more directly, it just boils down to $a^2 \equiv a \pmod{2a}$. This follows since $a (a-1) \equiv 0 \pmod{2a}$ as the first term is a multiple of $a$ and the second term is a multiple of 2.

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