Notes and Queries, Number 204, September 24, 1853 / A Medium of Inter-communication for Literary Men, Artists, Antiquaries, Genealogists, etc.

by Various

a₀ + a₁ . 10 + a₂ . 10² + ... + a_n-1 . 10^n-1 + a_n . 10ⁿ

Then

a_n + a_n-1 . 10 + a_n-2 . 10² + ... + a₁ . 10^n-1 + a₀ . 10ⁿ

is "the same number written backwards." The difference is—

(a_n - a₀)(10ⁿ - 1) + (a_n-1 - a₁)(10^n-2 - 1) . 10 + ...

+ (a_n/2+1 - a_n/2-1)(10²-1) . 10^n/2-1 if n be even, but

+ (a_(n+1)/2 - a_(n-1)/2)(10-1) . 10^(n-1)/2 if n be odd.

And every term of this difference, as involving a factor of the form (1 - 10ⁿ), is divisible by 9; and therefore the difference is divisible by 9.

C. Mansfield Ingleby.

Birmingham.