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Suppose that we are given: $$1124 \cdot 5097 \equiv x \mod 5693$$

Then $x = 1870$ since $(1124 \cdot 5097) -x = 5693k$ if $k=1006$.

Is this correct? I'm not exactly sure how to think about these types of problems. It seems that I could simply do $$(1124 \cdot 5097)\mod 5693 = 1870$$ If someone could please explain how to approach these types of problems, that would be greatly appreciated.

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In this specific case, you can just multiply the numbers out, then find the remainder. If, on the other hand, you had $a*x\equiv b\mod c$, you would need to find $a^{-1}\mod c$ and multiply both sides by it.

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