Suppose $n$ is a positive integer. Using induction, prove that there are unique integers $a\geq 0$ and $k>0$ such that $n=(3^a)\cdot k$ and $k$ is not divisible by $3$.
Note: I have already proven the base step of $P(1)$ and have set the induction hypothesis (I.H.) to be $P(t): t=(3^a)\cdot k$ [I used $t$ here instead of $k$ since $k$ already exists in the equation]. I am to the point of proving $P(t+1)$ but am unsure of whether or not this means $t+1=(3^a)\cdot k$ or if it means $t+1=(3^a)\cdot k+1$? With the latter, couldn't you just subtract a $1$ and then be left with your I.H.? If it is the first equation, I'm not quite sure how this can be proved. The induction proofs I have done so far were all summation problems, so I could group sections together and set it equal to the I.H. which doesn't seem to be a possibility here.