# Divisibility by 9/Corollary

< Divisibility by 9(Redirected from Divisibility by 3)

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## Corollary to Divisibility 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 $3$

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

## Proof

From Divisibility by 9 we have that:

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

- $a_0 + a_1 + \ldots + a_n$ is divisible by $3^2$.

So:

\(\displaystyle \paren {a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n}\) | \(\equiv\) | \(\displaystyle \paren {a_0 + a_1 + a_2 + \cdots + a_n}\) | \(\displaystyle \pmod {3^2}\) | ||||||||||

\(\displaystyle \leadstoandfrom \ \ \) | \(\displaystyle \paren {a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n}\) | \(\equiv\) | \(\displaystyle \paren {a_0 + a_1 + a_2 + \cdots + a_n}\) | \(\displaystyle \pmod 3\) | Congruence by Divisor of Modulus |

$\blacksquare$

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 2.6$. Algebra of congruences: Example $41$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $3$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $3$