2026/057) Find the smallest positive integer n such that $3^5 | 10^n-1$

Before finding the n let us make a few obsevations

We have for any n $9 | 10^n-1$

That is   $3^2 | 10^n-1$

And as $ 3| 111$ or $3 | \frac{10^3-1}{9}$

Hence $3^3 | 10^3-1$

Now we need to move from $3^3$ to $3^5$

999 is divisible by 27

Appending another 999 to the left shall give 999999 and it is divisible by 27 appending 9 or 99 does not make if divisible by 27

But appending 999 to the left is adding $999 * 10^3$

As $243 = 27 *9$ we need to keep a-dding until the  

Let us keep appending say n terms

We get $999(1+ 10^3 + 10^6 + \cdots 10^{3n})$

For this to be divisible by 243 as 999 is divisible by 27  $1+ 10^3 + 10^6 + \cdots 10^{3n})$ must be divisible by 9.

Each term of   $1+ 10^3 + 10^6 + \cdots 10^{3n})$ divided by 9 leaves a remainder 1 so there should by 8 terms and so n= 24

Then we get the number = $999(1+10^3 + \cdots 10^24)$

$= (10^3-1)(1+10^3 + \cdots 10^24)= 10^27-1$

so n= 27