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Mathematics Educators.

I'm having a difficult time understanding modular arithmetic (particularly its applications for proof writing). I understand very well the basics. I know that $a\equiv b \pmod n$ means that $a/n$ and $b/n$ produce the same remainder, thus I can verify this to be true if and only if $n|a-b$. I understand the basic properties of this and how to prove these properties, such as $a\equiv b \pmod n \implies ac\equiv bc \pmod n$.

What I'm struggling with specifically is knowing how to apply this logic to various proof problems. Using congruencies to solve for $x$ in $y=5x+11 \pmod{26}$, using congruencies to show that for odd $n$ that does not divide $3$, $n^2\equiv 1\pmod{24}$, etc. These are the kind of problems I'm having difficulty with.

Are there any free or cheap resources where I can practice with congruencies and have the full solution available afterwards to test myself? My course textbook has not been very helpful.

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    $\begingroup$ It may help you to have a phrase to google: "Diophantine equation" The Chinese Remainder Theorem also often comes in handy. So does general trickery, like saying that an odd $n$ not divisible by 3 means $n=1,5$ mod 6 and so $n=1,5,1+6,5+6,1+2\times 6, 5+2\times 6, 1+3\times 6, 5+3\times 6$ mod 24. $\endgroup$ – Adam Mar 31 at 13:29

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