Does Modular Arithmetic Prove a Prime Divides Infinite Repetitive Digit Numbers?

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Homework Help Overview

The discussion revolves around proving that a prime number \( p \) (other than 2 or 5) divides infinitely many integers of the forms 9, 99, 999, etc., and 1, 11, 111, etc. The participants are exploring the use of modular arithmetic to establish this relationship.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster inquires about using modular arithmetic to prove the divisibility of certain integers by a prime \( p \). Some participants are questioning how to start the proof and whether different powers of 10 can be shown to be congruent modulo \( p \).

Discussion Status

The discussion is in an early stage, with participants expressing uncertainty about how to begin the proof. There is an indication that some foundational concepts related to modular arithmetic are being revisited, particularly regarding congruences.

Contextual Notes

Participants are focusing on primes other than 2 and 5, which may influence the nature of the integers being considered. The problem appears to be framed within the constraints of a homework assignment, emphasizing the need for a proof rather than a solution.

keityo
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if p is any prime other than 2 or 5, prove that p divides infinitely many of the integers 9, 99, 999, 9999, ... If p is any prime other than 2 or 5, prove that p divides infinitely many of the integers 1, 11, 111, 1111, ...

Is there a way to do this problem using modular arithmetic? Thanks
 
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Yes.
 
I am stuck. Could you help me?
 
Where are you stuck? What have you done so far?
 
On getting started.
 
keityo said:
On getting started.

Can you prove that at least 2 different powers of 10 have to be equal to each other mod p?
 
Last edited:

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