When $555_{10}$ is expressed in this base, it has 4 digits, in the form ABAB, where A and B are two different digits. What base is it?

When $555_{10}$ is expressed in this base, it has 4 digits, in the form ABAB, where A and B are two different digits. What base is it?

Let the base be X

$$\\A*X^3+B*X^2+A*X+B=555\\\\
AX^3+BX^2+AX+B=555\\\\
X^2(AX+B)+(AX+B)=555\\\\
(X^2+1)(AX+B)=555\\\\
So\;\;X^2+1\;\;$must be a factor of 555$\\\\$$

$${factor}{\left({\mathtt{555}}\right)} = {\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{5}}{\mathtt{\,\times\,}}{\mathtt{37}}$$

factors of 555 are   3,5,37,15,111,185,555

5  and 37 would both work so far

5 would  be base 2 And base 2 is to little,  37 would give base 6 (X=6) so that is probably correct.

Lets see

$$\\(X^2+1)(AX+B)=555\\\\
(6^2+1)(6A+B)=555\\\\
37(6A+B)=555\\\\
6A+B=15\\\\
If A=1 B=8 \qquad $no good A and B cannot be more than 5$\\\\
If A=2 B=3 \qquad $Great$\\\\
$so our number is $ 2323_6$$

Check

$${\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{6}}}^{{\mathtt{3}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}}{\mathtt{\,\times\,}}{{\mathtt{6}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{6}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}} = {\mathtt{555}}$$

And that is excellent

So               $$555_{10}=2323_6$$

That's a nice approach and nice thinking, Melody....

Thanks Chris :)