Nonary - Futility Closet

Nonary

Take a whole number, reverse the order of its digits, and subtract one from the other. The difference will always be evenly divisible by 9.

Does this remain true if we just scramble the digits of the first number, rather than reversing them?

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Yes. Suppose we scramble 27910 to get 19702: 27910 = 2 × 10⁴ + 7 × 10³ + 9 × 10² + 1 × 10¹ + 0 × 10⁰ 19702 = 1 × 10⁴ + 9 × 10³ + 7 × 10² + 0 × 10¹ + 2 × 10⁰ Now collect the terms of 27910 – 19702 that have the same coefficients: 2 × (10⁴ – 10⁰) + 7 × (10³ – 10²) + 9 × (10² – 10³) + 1 × (10¹ – 10⁴) + 0 × (10⁰ – 10¹) Each term has a factor that’s either 10^m(10ⁿ – 1) or -10^m(10ⁿ – 1) for some integers m, n. So each term, and therefore the sum, is divisible by 9. From Keith Kendig, Sink or Float?: Thought Problems in Math and Physics, 2008.