Divisibility by 9

Theorem

A number expressed in decimal notation is divisible by $9$ if and only if the sum of its digits is divisible by $9$.

That is:

$N = \sqbrk {a_0 a_1 a_2 \ldots a_n}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $9$

if and only if:

$a_0 + a_1 + \cdots + a_n$ is divisible by $9$.

A number expressed in decimal notation is divisible by $3$ if and only if the sum of its digits is divisible by $3$.

That is:

$N = \sqbrk {a_0 a_1 a_2 \ldots a_n}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $3$

if and only if:

$a_0 + a_1 + \ldots + a_n$ is divisible by $3$.

Let $N$ be divisible by $9$.

Then:

$\ds N$

$\equiv$

$\ds 0 \pmod 9$

$\ds \leadstoandfrom \ \ $

$\ds a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$

$\equiv$

$\ds 0 \pmod 9$

$\ds \leadstoandfrom \ \ $

$\ds a_0 + a_1 1 + a_2 1^2 + \cdots + a_n 1^n$

$\equiv$

$\ds 0 \pmod 9$

as $10 \equiv 1 \pmod 9$

$\ds \leadstoandfrom \ \ $

$\ds a_0 + a_1 + \cdots + a_n$

$\equiv$

$\ds 0 \pmod 9$

$\blacksquare$

$\blacksquare$

1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Some Properties of $\Z$: Exercise $2.8$
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.3$: Congruences: Exercise $9$
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $9$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $9$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): divisible
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): divisible