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Under the identities section in this Wikipedia page they list the distributive property for multiplication. However it doesn't hold true for fractions. Is there a list of modulo properties when fractions (particularly irrational fractions) are involved?

Listed rule

ab % x == ((a % x) * (b % x)) % x

Counter Example

(4 * .002) % 1 != 0
mikeLundquist
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  • Does this post help? https://math.stackexchange.com/questions/1914822/fractions-in-modular-arithmetic/1914834 – Striker Sep 27 '21 at 13:54
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    The normal rule that I was taught is that for $r,s,t \in \Bbb{Z}, ~ t\neq 0,~$ you have that $~r \equiv s \pmod{t} ~\iff ~t ~| ~(r - s) ~\iff ~\exists ~k \in \Bbb{Z} ~\text{such that} ~(t \times k) = (r - s).$ Personally, I see no problem with extending this idea to non-integer $~r,s,t$. For example, $2\pi \equiv -\pi/2 \pmod{\pi/4}$ because $(\pi/4) \times 10 = [2\pi - (-\pi/2)].$ – user2661923 Sep 27 '21 at 13:54

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