Divisibility by 9/Corollary

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 $3$

if and only if:

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

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 $3^2$

if and only if:

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

So:

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

$\equiv$

$\ds \paren {a_0 + a_1 + a_2 + \cdots + a_n}$

$\ds \pmod {3^2}$

$\ds \leadstoandfrom \ \ $

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

$\equiv$

$\ds \paren {a_0 + a_1 + a_2 + \cdots + a_n}$

$\ds \pmod 3$

$\blacksquare$

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): $3$
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