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

[keityo]

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

 

[Hurkyl]

Yes.

 

[keityo]

I am stuck. Could you help me?

 

[Hurkyl]

Where are you stuck? What have you done so far?

 

[keityo]

On getting started.

 

[ramsey2879]

On getting started.

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