# Multiple of 999 can be Split into Groups of 3 Digits which Add to 999

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## Contents

## Theorem

Let $n$ be a (strictly positive) integer multiple of $999$.

Then it is possible to split $n$ up into groups of $3$ digits, counting from the unit position, to form $3$-digit integers which add up to $999$.

### Examples

\(\displaystyle 4 \times 999\) | \(=\) | \(\displaystyle 3996\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 3 + 996\) | \(=\) | \(\displaystyle 999\) |

\(\displaystyle 15 \times 999\) | \(=\) | \(\displaystyle 14 \, 985\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 14 + 985\) | \(=\) | \(\displaystyle 999\) |

\(\displaystyle 47 \times 999\) | \(=\) | \(\displaystyle 46 \, 953\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 46 + 953\) | \(=\) | \(\displaystyle 999\) |

\(\displaystyle 57 \times 999\) | \(=\) | \(\displaystyle 56 \, 943\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 56 + 943\) | \(=\) | \(\displaystyle 999\) |

## Proof

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $999$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $999$