Application: Divisibility criteria revisited

Let

$\displaystyle a=a_k10^k+...a_1 10+a_0, $

where $ 0\leq a_i\leq 9$ are the digits. The following congruence conditions generalize some of the criteria given in §1.7.1.

mod 2
: $ a \equiv a_0\ ({\rm mod}\ 2)$.

mod 3
: $ a \equiv (a_0+a_1+...+a_k)\ ({\rm mod}\ 3)$.

mod 4
: $ a \equiv (a_0+a_1 10)\ ({\rm mod}\ 4)$.

mod 5
: $ a \equiv a_0\ ({\rm mod}\ 5)$.

mod 8
: $ a \equiv (a_0+10a_1+100a_2)\ ({\rm mod}\ 8)$.

mod 9
: $ a \equiv (a_0+a_1+...+a_k)\ ({\rm mod}\ 9)$.

For example, $ 11116\equiv 1 \ ({\rm mod}\ 9)$.

mod 10
: $ a \equiv a_0\ ({\rm mod}\ 10)$.

There are also conditions for $ 7$, $ 11$, and $ 13$ but they are left as exercises.


David Joyner 2007-09-03