Show that there are no squares included in the sequences

Question

Show that there are no squares included in in the sequences

(11, 111, 1111, 11111, ....)

(22, 222, 2222, 22222, ....)

(33, 333, 3333, 33333, ....)

and so on and so forth for all numbers $1 \rightarrow 9$

I was thinking about showing that because 11 is not square then some modulo 11, or 111, etc... is not square. Something like that, but I'm getting all these definitions mixed up in my head.

Any help to kickstart me in the right direction?

Answer 1

As I have commented $\displaystyle\underbrace{11\cdots11}_{n(\ge 2)\text{ digits}}\equiv11\pmod4\equiv-1\pmod 4$

But, any integer $a\equiv0,\pm1,2\pmod4\implies a^2\equiv0,1\pmod4$

So, $\displaystyle\underbrace{11\cdots11}_{n(\ge 2)\text{ digits}}$ can not be perfect Sqaure

Now, $\displaystyle99\cdots99$ and $\displaystyle44\cdots44$ will be square if and only if $\displaystyle11\cdots11$ is perfect square

$\displaystyle22\cdots22;66\cdots66;88\cdots88$ will not be perfect square as the highest power of $2$ are odd

For the same reason, $\displaystyle55\cdots55;77\cdots77$ will not be perfect square