The 4 digit base $6$ number $abcd$ with $a>0$ and $d$ odd is a perfect square. 
The 4 digit  base $6$ number $abcd$ with $a>0$ and $d$ odd  is a perfect square. List all possible values of $c$. (The letters are the digits of the base 6 number.)

I've rewritten $abcd_6$ into $216a + 36b + 6c + d$, and I know that $d$ can be $1,3,5$, but now I'm stuck. I guess I could start plugging in some numbers since $a,b,c,d \le 5$, but I feel like there would be a more efficient way. 
 A: The odd perfect squares in the interval $[217,1295]$ where we are interested are $225,289,361,441,529,625,729,841,961,1089,1225$. Reducing these modulo $36$ we get $9,1,1,9,25,13,9,13,25,9,1$. Note these are exactly the odd quadratic residues modulo $36$ but here we have shown that they all crop up in our desired interval. The quotients of these numbers when divided by $6$ gives the possible values of $c$: $0,1,2,4$.
A: Work modulo $36$, test only the odd numbers $15$ through $35$:
$^\text{(we may omit $1$ through $13$, since these would not satisfy $a>0$)}$
$$\,$$
$$(15)^2 \equiv 9 \pmod{36} $$
$$(17)^2 \equiv 1 \pmod{36} $$
$$(19)^2 \equiv 1 \pmod{36}$$
$$(21)^2 \equiv 9 \pmod{36} $$
$$(23)^2 \equiv 25 \pmod{36}$$
$$(25)^2 \equiv 13 \pmod{36} $$
$$(27)^2 \equiv 9 \pmod{36} $$
$$(29)^2 \equiv 13 \pmod{36} $$
$$(31)^2 \equiv 25 \pmod{36}$$
$$(33)^2 \equiv 9 \pmod{36} $$
$$(35)^2 \equiv 1 \pmod{36} $$
$$\,$$
The possible remainders for odd perfect squares, modulo $36$, are $1$, $9$, $13$, and $25$. 

In base $6$, these are equivalent to the last two digits being $$01_6,\,13_6,\,21_6,\,41_6$$
In particular, $$\boxed{\,c \, \in \, \left\{0,1,2,4\right\}\,\,}$$
A: To look at the last two digits in base $6$, we want to look at $4k^2+4k+1$ mod $36$. Since $k^2+k$ mod $9$ repeats with period $9$, $4k^2+4k+1$ mod $36$ also repeats with period $9$. Therefore, the the last two digits of odd squares in base $6$ also repeat with period $9$.
Perhaps more concisely,
$$
\begin{align}
(2(k+9)+1)^2
&=(2k+19)^2\\
&=4k^2+76k+361\\
&=\left(4k^2+4k+1\right)+(72k+360)\\
&=(2k+1)^2+36(2k+10)
\end{align}
$$
Thus, $(2k+1)^2$ repeats mod $36$ with period $9$. That is, the last two digits of $(2k+1)^2$ in base $6$ repeat with period $9$.
Thus, all the possible values of the $6^1$ digit in base $6$ are given in red:
$$
\begin{array}{c|c|r}
k&(2k+1)^2&\text{base $6$}\\
\hline
0&1&\color{#C00}{0}1\\
1&9&\color{#C00}{1}3\\
2&25&\color{#C00}{4}1\\
3&49&1\color{#C00}{2}1\\
4&81&2\color{#C00}{1}3\\
5&121&3\color{#C00}{2}1\\
6&169&4\color{#C00}{4}1\\
7&225&10\color{#C00}{1}3\\
8&289&12\color{#C00}{0}1\\
\end{array}
$$
These are all the possible values for any odd square in base $6$, in particular from $15^2$ to $35^2$.
