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  • $\begingroup$ I think the first two steps are are not needed since your proof is not by contradiction and you don't use the fact that the remainders not all different. $\endgroup$ Commented Jul 23, 2023 at 22:23
  • $\begingroup$ Yup, I had a different idea in mind initially, so later deleted some stuff, but forgot to remove all those parts. $\endgroup$ Commented Jul 23, 2023 at 22:27