1 x 8 + 1 = 9 12 x 8 + 1 = 98 123 x 8 + 1 = 987 1234 x 8 + 1 = 9876 12345 x 8 + 1 = 98765

The proof is via this reddit comment.

Theorem:

[12...n] x (b-2) + n = [(b-1)(b-2)...(b-n)]

Notation `[12..n]`

means a number written out as such.

[12...n] x (b-2) + n = [12...n] x b - [12...(n-1)n] - [12...n] + n (1) = [12...n0] - [12...(n-1)0] - [12...n] (2) = [11...10] - [12...n] (3) = [(b-1)(b-2)...(b-n)] (4)

*a × (b-2)*is the same as*a × b - a - a*.- Multiplying a digit in base
*b*by*b*shifts it left by 1, so*[12…n] × b*is*[12…n0]*. Also note that subtracting*n*from*[12…n]*will give you*[12…0]*. The*n-1*was added to clarify. - As a concrete example:
*1230-120 = 1110* - As a concrete example:
*1110-123 = 987*

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