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Can anyone help me to understand what to do with the subtraction or addition on the LHS of a congruence? I found this question -

Solving congruences involving addition for CRT

But I don't understand how the answer goes from $$7j+6\equiv 4 \pmod{5}$$ to $$7j\equiv −2\equiv 3 \pmod{5}$$ from subtracting 6, and then something happens and things become familiar where $2j\equiv 3\pmod{5}$.

What rules are being applied here? What is being subtracted, and why does the $4 \pmod{5}$ turn to $3\pmod{5}$?

I could solve $2j\equiv 3\pmod{5}$, but I don't know how we got there from $7j+6\equiv 4 \pmod{5}.$

I just don't know what to do about $b$ given $ax + b = c\pmod{n}.$

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    $7j+\color{blue}6\equiv4\pmod5\iff 7j\equiv4-\color{blue}6=-2\pmod5\iff 7j\equiv3\pmod5$ because $-2\equiv3\pmod5$ – J. W. Tanner Nov 03 '21 at 01:16
  • Thank you that was very helpful! That makes a lot more sense now. I seen that `4 - 6 = -2` but I wasn't sure what to do next. Where does the `2` in `2j≡3mod5` come from? is it just the absolute value from `-2`? – null Nov 03 '21 at 01:24
  • $7j\equiv2j\pmod5$ because $7\equiv2\pmod5$; glad to help – J. W. Tanner Nov 03 '21 at 01:31
  • Awesome! I very much appreciate the help. If you want to post as an answer I'll accept. – null Nov 03 '21 at 01:34
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    I would like to post an answer, but I'm afraid that someone will say that this question is similar to a previous one, and then I could be suspended for answering a duplicate. Anyways, welcome to Mathematics Stack Exchange. – J. W. Tanner Nov 03 '21 at 01:40
  • They add $-6$ to both sides fo the congruence using the [Congruence Sum Rule](https://math.stackexchange.com/a/879262/242) in the linked dupe, i.e. $\, A\equiv B,\ {-}6\equiv -6\Rightarrow A-6\equiv B-6.\ $ Congruences are generalized equations, and like equations they are preserved by adding (or multiplying) both sides by equal numbers (says the Sum & Product Rules). – Bill Dubuque Nov 03 '21 at 01:42
  • SImilarly $\ 7\equiv 2\Rightarrow 7j\equiv 2j\ $ by the [Congruence Product Rule](https://math.stackexchange.com/a/879262/242). Thus by induction, the congruence rules imply that we can replace arguments of sums and products (but not expts!) by any congruent argument and we will obtain a congruent result - as explained [here](https://math.stackexchange.com/a/3266937/242) – Bill Dubuque Nov 03 '21 at 01:52
  • Typo alert: "by equal numbers" should be "by congruent numbers" in the 2nd last comment. – Bill Dubuque Nov 03 '21 at 07:57

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