<s>[INST] ${3^n\choose k}$ is divisible by $3$? How can I prove that ${3^n\choose k}$ is divisible by $3$ for all positive integer values of $n$? (where $k$ is any positive integer smaller than $3^n$) Can you use induction? Thanks.
 [/INST] Looking at the fractional expansion of the binomial coefficient, you can cancel out factors of 3 in the $(3n+1)^\text{th}$ value in the numerator with the $(3n)^\text{th}$ value in the denominator, for example:
$${3^3 \choose k} = \begin{array} {c}
27 & 26 & 25 & \color{red}{24} & 23 & 22 & \color{blue}{21} & 20 & 19 & \color{green}{18} & 17 & 16 & \color{orange}{15} & \dots & 27 - (k - 1)  \\ \hline
1 & 2 & \color{red}{3} & 4 & 5 & \color{blue}{6} & 7 & 8 & \color{green}{9} & 10 & 11 & \color{orange}{12} & 13 & \dots & k
\end{array}$$
So you get $\pi_3(\text{Numerator}) - \pi_3(\text{Denominator}) = \pi_3(3^n) - \pi_3(k) > 0$.
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