How do I check 12321 ≡ 111 (mod 3) is true without applying the definition?

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How do I check 12321 ≡ 111 (mod 3) is true without applying the definition?

Mathematics
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12321 = 111^2 btw
as far as i see both are divisible \(3\)
yeah, but that's more like by inspection. I need to use properties of congruences

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Other answers:

12321 and 111 are divisible by 3 is sufficient
\(12321\equiv 0 \pmod {3}\) and \(111\equiv 0\pmod{3}\) \(\implies 12321 \equiv 111\pmod{3}\)
by transitive property?
Yep \(a\equiv c\pmod{n}\) and \(c\equiv b\pmod{n}\) \(\implies a\equiv b\pmod{n}\)
ah yes. I guess this was too easy. Let me pick a different problem. 12345678987654321 ≡ 0 (mod 12345678) ^^
It's according to wolfram alpha, But 12345678987654321 ≡ 0 (mod 12345679) is true though
it's *false* according...
12345678987654321 / 12345679 = 999,999,999
This problem seems challenging but it's in the section where it covers algebraic properties of congruences (addition,subtract and such...). I'm just assuming the exercise is meant for us to apply those properties but it doesn't seem like it.
Maybe the exercises are meant to check the readers' understanding of the definition. Who knows.

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