<s>[INST] Find the remainder when $3^{29}$ is divided by $12$. 
Find the remainder when $3^{29}$ is divided by $12$. 
  a) $2$ ; b) $3$ ; c) $7$ ; d) $9$ ; e) $12$

Since $\dfrac{3^{29}}{12} = \dfrac{3^{28}}{4} = \dfrac{9^{14}}{4}$, and $9 \equiv 1 \mod 4$, I thought I could do $9^{14} \equiv 1^{14} \mod 4$, so that the answer is $1$. However, that is not one of the answer choices... Where did I go wrong?
 [/INST] $$3^2\equiv1\pmod4,3^{2n}=(3^2)^n\equiv1^n$$
$$3^{2n+1}\equiv3\pmod{4\cdot3}$$
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