Divisibility by 9/Corollary
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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:
\(\ds \paren {a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n}\) | \(\equiv\) | \(\ds \paren {a_0 + a_1 + a_2 + \cdots + a_n}\) | \(\ds \pmod {3^2}\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \paren {a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n}\) | \(\equiv\) | \(\ds \paren {a_0 + a_1 + a_2 + \cdots + a_n}\) | \(\ds \pmod 3\) | 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): $3$
- 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