School-level problem on divisibility

Question

I encountered the problem to show that there is an integer of the form $11111\ldots 11$ divisible by $2021$. It is easy to show that there is a number of the form $111 \ldots 11 \cdot 10^k$ divisible by $2021$. But I can't use the unique factorisation to get rid of $10^k$ (the problem says no unique factorisation). So how can I prove that?

Answer 1

$18189=9\times 2021$ is relatively prime with $10$.

So there exists $k$ such that $10^k\equiv 1\pmod{18189}$

$k=966=(2)(3)(7)(23)$ can be tested from divisors of Euler totient function $\phi(18189)=(2)^3(3)^2(7)(23)$.

Thus the repunit $\frac 19(10^k-1)$ is divisible by 2021.

Answer 2

For every natural number $m$($m$ is co prime to $2$ and $5$), there is a respective string of $1$ , that is $1111...11$ which is divisible by $m$.

Proof: For, some number $n$, we know $n|10^{\phi(n)}-1$, from Euler's theorem.

We can write any number which is a string of $1$s$(11111....)$ in base $10$ as $1+10^{1}+10^{2}+....10^{a-1}=\frac{10^{a}-1}{9}$

Now, if $\text{gcd}(n,9)=1,$ we can put $a=k\phi(n)$ without any hesitation. Because the $9$ in denominator will not take out any factor common with $n$.

But, if $\text{gcd}(n, 9)≠1$, then $n=3^{l}p$. And we can easily show that the power of $3$ in ${10^{k\phi(n)}-1}$ is more than $l+1$ for all $k>k_0$ for some $k_0>1.$

And this way we have proved the statement to be true. And even by this algorithm you can make such strings.

There may be more other strings of $1$ for few $n$s. I mean other than this algorithm.