<s>[INST] Finding the correct statements about $(5+2\sqrt{6})^{2n+1} = S + t$ with $S$ integer and $0 \leq t < 1$ Problem:

If $n$ is a positive integer and $(5+2\sqrt{6})^{2n+1} = S + t$, where $S$ is an integer and $0 \leq t < 1$, then
(a) $S$ is an odd integer
(b) $S + 1$ is not divisible by $9$
(c) The integer next above $(5+2\sqrt{6})^{2n+1}$ is divisible by $10$
(d) $S-1 = \frac{t}{t-1}$

Please guide me how to proceed with this question...
 [/INST] Hint:  note that $(5+2\sqrt 6)^{2n+1}+(5-2\sqrt 6)^{2n+1}$ is an integer, and $(5-2 \sqrt 6) \lt 1$, so a power of it will be small and positive.
</s>