For any k-digit block repeated r times, the repetitious-number trick works by dividing by the prime factors of 1 + 10^k + ... + 10^{k(r-1)}, which recovers the original block.
G UTKOVSKIY , Sequence A261544 in The On-Line Encyclopedia of Integer Se- quences (Aug 24, 2015) , published electronically at https://oeis.org
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A Recreational Application of Two Integer Sequences and the Generalized Repetitious Number Puzzle
For any k-digit block repeated r times, the repetitious-number trick works by dividing by the prime factors of 1 + 10^k + ... + 10^{k(r-1)}, which recovers the original block.