One can check arithmetic in base 10 by casting out nines, and similarly computing remainders by alternating sums for 11 and the beautiful 1001=7*11*13. (So 11 is repeated effort.) We can also get 999=37*27 and prime 101. Is base 10 especially nice in its ease of checking arithmetic? We are probably looking for large least common multiple of b^n plus/minus 1, over small n.

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