Find the three smallest positive integers $K$ (two digits or greater) with the following properties:
1) $K=\frac{(n)(n+1)}{2}$ for some $n$
2) Each digit of $K$ is the same.
I was able to find the three $K's$ which are $55,66,666$ However, I am looking for a more systematic way. I was able to find the three numbers by testing $n$ I got the following:
For:
$55: n=10 $
$66: n=11 $
$666: n=36$
I am thinking about how can someone solve this problem without a calculator. I believe this problem is written to use a certain idea but I can't seem to find it. Does anyone see it?
As for my research, I've been reading about triangular numbers with the same digits. Someone suggested that I should $(n)$ and $(n+1)$ in prime factorization form.
I noticed that you can't get $6666$
If you try $n=115$, you get $6670$. So $6666$ is not possible.
So basically, I am looking for strategies in how to approach this problem. Any ideas?