_a__{_n_} + _a__{_n_-1} . 10 + _a__{_n_-2} . 10^2 + ... + _a__1 .
10^{_n_-1} + _a__0 . 10^{_n_}
is "the same number written backwards." The difference is--
(_a__{_n_} - _a__0)(10^{_n_} - 1) + (_a__{_n_-1} - _a__1)(10^{_n_-2} - 1)
. 10 + ...
+ (_a__{_n_/2+1} - _a__{_n_/2-1})(10^2-1) . 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 -
10^{_n_}), is divisible by 9; and therefore the difference is divisible by
9.