Casting Out Nines

An elementary check of a Multiplication which makes use of the Congruence $10^n\equiv 1$ (mod 9) for $n\geq 2$. From this Congruence, a Multiplication $ab=c$ must give

$$a\equiv \sum a_i = a^*$$

$$b\equiv \sum b_i = b^*$$

$$c\equiv \sum c_i = c^*,$$

so $ab \equiv a^*b^*$ must be $\equiv c^*$ (mod 9). Casting out nines is sometimes also called ``the Hindu Check.''

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 28-29, 1996.