"Casting out nines" is an elementary check of a multiplication which makes use of the congruence 10^n congruent 1 (mod 9). Let decimal numbers be written a = a_n ...a_2 a_1 a_0, b = b_n ...b_2 b_1 b_0, and their product be c = c_n ...c_2 c_1 c_0. Let the sums of the digits of these numbers be a^*, b^*, and c^*. Then a congruent a^* (mod 9), b congruent b^* (mod 9), and c congruent c^* (mod 9). Furthermore a b congruent a^* b^* (mod 9), so c congruent c^* (mod 9). So if c and a^* b^* are incongruent (mod 9), the multiplication has been done incorrectly.
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