Modular Arithmetic Simplify

Question

Congruence is an equivalence relation so transitive, so $\,x^2\equiv 79,\ 79\equiv 2\Rightarrow x^2\equiv 2,\,$ see the linked dupe.

I was wondering how in number theory we are able to determine how to quickly simplify modular congruences. So for example if I had the equation

$$x^{2} \equiv 79 \pmod 7.$$

is the same as

$$x^{2} \equiv 2 \pmod 7$$

How do I know quickly for any formula to find the smaller and equivalent congruence? I know that 77 is a factor and 77+2 is 79, but how is that the same as 2 mod 7? whats the point of having the larger number?

When you said that $77$ is a factor, did you mean that $77$ is a multiple (of $7$)? — J. W. Tanner, May 14 '23 at 19:38
You write $79=77+2$ as if there's more to see, but you're already there. Maybe it's more illuminating to write $79 = 7(11)+2$? This is literally what it means for $79 \equiv 2 \pmod 7$ since $79$ has a remainder of $2$ when divided by $7$. In terms of what's the point, this is in a sense the same idea as the point of writing $\frac{3}{4} = \frac{6}{8}$, they're just equivalent expressions. — Irving Rabin, May 14 '23 at 19:38
Congruence is an equivalence relation so transitive, so $,x^2\equiv 79,\ 79\equiv 2\Rightarrow x^2\equiv 2,,$ see the first linked dupe. That $,79\equiv 2,$ is a special case of $,a\equiv (a\bmod n)\pmod{!n},,$ see the 2nd dupe. — Bill Dubuque, May 14 '23 at 20:06
OP asks, "What's the point of having the larger number?" One point is to force the solver to understand how to manipulate congruences. But in general, if we didn't just make up the example for teaching purposes, it might arise in other ways. For example, one could ask a general question about what prime numbers are congruent to squares with respect to various moduli, and in particular wonder if the $22$nd prime number was a perfect square $\bmod 7$. Then you would set up the equation in the question and solve it. — Keith Backman, May 14 '23 at 20:16

Modular Arithmetic Simplify

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