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  • @FelipeC : if you're using the euclidean division approach, then whether you stayed up for 6 hours or 30 hours will still yield -18 (mod -24), which really provides you with no helpful information, since staying up for 6 versus 30 aren't remotely similar to each other. Commented Apr 3, 2024 at 17:19
  • @RAREKpopManifesto in both cases what remains of the day is 18 hours, so yes it provides helpful information. Commented Apr 4, 2024 at 10:28
  • that's a false equivalence cuz it involves asking a different question altogether - being up 30 hours mean the -18 refers to "how many hours you're short of 2 full days", while being up 6 hours mean -18 is "how many hours you're short of 1 full day". To be asking the same question of "how many hours you're short of 2 full days", the congruent value needs to be -42. The only congruence of your approach is "how many hours it's short of the next midnight after you went to bed" - a totally different question altogether. Commented Apr 8, 2024 at 4:26
  • @RAREKpopManifesto you misunderstand the abstract algebra. There is no 30 in integers modulo -24 (ℤ/-24ℤ). The members of this set are themselves a set, so the mathematical object we are talking about is {54, 30, 6, -18, -42}. 6 and 30 are not just equivalent, they are exactly the same mathematical object. Commented Apr 9, 2024 at 19:06