# Digital Root of 3 Consecutive Numbers ending in Multiple of 3

Jump to navigation
Jump to search

## Contents

## Theorem

Let $n$, $n + 1$ and $n + 2$ be positive integers such that $n + 2$ is a multiple of $3$.

Let $m = n + \paren {n + 1} + \paren {n + 2}$.

Then the digital root of $m$ is $6$.

## Proof

Let $n + 2$ be expressed as $3 r$ for some positive integer $r$.

Then:

\(\displaystyle m\) | \(=\) | \(\displaystyle \paren {3 r - 2} + \paren {3 r - 1} + 3 r\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 9 r - 3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 9 \paren {r - 1} + 6\) |

The result follows from Digital Root of Number equals its Excess over Multiple of 9.

$\blacksquare$

## Historical Note

According to David Wells in his *Curious and Interesting Numbers, 2nd ed.* of $1997$, this result is due to Iamblichus Chalcidensis.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $6$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $6$