prove that there are infinitely many numbers of the form $x = 111....1$ such that $31\mid x$

Question

I need to prove that there are infinitely many numbers of the form $x = 111....1$ such that $31\mid x$

what I tried - I wrote x as

$\sum_0^{n-1} 10^i$

I know that $(10,31) = 1 $

now I'm stuck .. any help will be appreciated

Answer 1

$$\underbrace{11\cdots11}_{n\text{ digits}}=\dfrac{10^n-1}9$$

Using Fermat's Little Theorem, $$10^{31-1}\equiv1\pmod{31}\iff31\mid(10^{30}-1)$$

Now, $(10-1)\mid(10^m-1)$ for integer $m\ge0$

As $(3,31)=1$ $$31\mid\dfrac{10^{30}-1}9$$

Finally, $10^r-1$ will be divisible by $10^{30}-1$ if $30\mid r$

Answer 2

Hint $\ $ By a simple Pigeonhole argument, if $\rm\:n\:$ is coprime to $10\,$ then every integer with at least $\rm\:n\:$ digits all $\ne 0$ has a contiguous digit subsequence that forms an integer $\ne 0\,$ divisible by $\rm\:n.\:$