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For example, "Solve the congruence (if possible), listing all the incongruent solutions:"

$$561x\equiv 3575\mod{1562}$$

I found $x\equiv 37+142t,\ 0\leq t\leq 10,\ t\in\mathbb{Z}$... There are 11 "incongruent solutions" because $(561,1562)=11$ and $11\mid 3575$... but what does "incongruent" mean?

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2 Answers

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Incongruent (in this case) means distinct modulo $1562$. For example, $1$ and $1561$ are incongruent modulo $1562$, but $1$ and $1563$ are not (rather, they are congruent modulo $1562$).

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if we consider $ x= 37,1599,3161....... $ then all are solution for x and that all are congruent each other because they are all 37 more than a multiple of 1562. so they all belongs to a same congruence class usually denoted by [37]

also satisfying the congruence are $ x=179, 1741, 3303 $ but these solutions,, belongs to the same congruence class, usually denoted by [179] are not congruent to any solution in class [37] mod 1562.

similar to the other 9 solutions

so x= 37, 179, ... are incongruence solutions because they belong to different congruence classes.

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