# divisibility of nine-numbers

We know that 9 is divisible by the prime number^{} 3 and that 99 by another prime number 11. If we study the divisibility other “nine-numbers” by primes, we can see that 999 is divisible by a greater prime number 37 and 9999 by 101 which also is a prime, and so on. Such observations may be generalised to the following

Proposition. For every positive odd prime $p$ except 5, there is a nine-number $999\mathrm{\dots}9$ divisible by $p$.

*Proof.* Let $p$ be a positive odd prime $\ne 5$. Let’s form the set of the integers

$9,\mathrm{\hspace{0.17em}99},\mathrm{\hspace{0.17em}999},\mathrm{\dots},\underset{p\mathrm{nines}}{\underset{\u23df}{99\mathrm{\dots}9}}.$ | (1) |

We make the antithesis that no one of these numbers is divisible by $p$. Therefore, their least nonnegative remainders modulo $p$ are some of the $p-1$ numbers

$1,\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3},\mathrm{\dots},p-1.$ | (2) |

Thus there are at least two of the numbers (1), say $a$ and $b$ ($$), having the same remainder. The difference $b-a$ then has the decadic of the form

$$b-a=\mathrm{\hspace{0.33em}999}\mathrm{\dots}9000\mathrm{\dots}0,$$ |

which comprises at least one 9 and one 0. Because of the equal remainders of $a$ and $b$, the difference is divisible by $p$. Since $b-a=999\mathrm{\dots}9\cdot 1000\mathrm{\dots}0$ and 2 and 5 are the only prime factors^{} of the latter factor (http://planetmath.org/Product), $p$ must divide the former factor $999\mathrm{\dots}9$ (cf. divisibility by prime). But this is one of the numbers (1), whence our antithesis is wrong. Consequently, at least one of (1) is divisible by $p$.

In other http://planetmath.org/node/3313positional digital systems, one can write propositions analogous to the above one concerning the decadic system, for example in the dyadic (a.k.a. digital system:

Proposition. For every odd prime $p$, there is a number $111\mathrm{\dots}{1}_{\mathrm{two}}$ divisible by $p$.

Title | divisibility of nine-numbers |
---|---|

Canonical name | DivisibilityOfNinenumbers |

Date of creation | 2013-03-22 19:04:43 |

Last modified on | 2013-03-22 19:04:43 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 11A63 |

Classification | msc 11A05 |