Application: Divisibility criteria revisited

Let

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

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.