Convert the following numbers into ternary system.
$$(17)_{10}$$
$$(89)_{10}$$
$$(121)_{10}$$
$$(3D)_{18}$$
Development:
We have to do all the possible integer divisions of the numbers by $$3$$:
$$\begin{eqnarray} &(17)_{10} \Rightarrow & 17 & |\underline{3} & & \\\\ & & \fbox{2} & 5 & |\underline{3} & \\\\ & & & \fbox{2} & \fbox{1} \end{eqnarray}$$
So that:
$$(17)_{10}=(122)_3$$
In the second case:
$$\begin{eqnarray} &(89)_{10} \Rightarrow & 89 & |\underline{3} & & & & \\\\ & & \fbox{2} & 29 & |\underline{3} & & & \\\\ & & & \fbox{2} & 9 & |\underline{3} & & \\\\ & & & & \fbox{0} & 3 & |\underline{3} \\\\ & & & & & \fbox{0} & \fbox{1} \end{eqnarray}$$
Then,
$$(89)_{10}=(10022)_3$$
In the third number of the exercise:
$$\begin{eqnarray} &(121)_{10} \Rightarrow & 121 & |\underline{3} & & & & \\\\ & & \fbox{1} & 40 & |\underline{3} & & & \\\\ & & & \fbox{1} & 13 & |\underline{3} & & \\\\ & & & & \fbox{1} & 4 & |\underline{3} \\\\ & & & & & \fbox{1} & \fbox{1} \end{eqnarray}$$
The equivalent is:
$$(121)_{10}=(11111)_3$$
Finally, in the last exercise it is necessary to combine the decomposition with the divisions, since the number is not in base $$10$$ but in base $$18$$.
First, into decimal:
$$(3D)_{18}=(3(13))_{18}=3\cdot18^1+13\cdot18^0=54+13=67$$
Now, to find the equivalent of $$67$$ in ternary, it is necessary to divide the above mentioned number successively by $$3$$:
$$\begin{eqnarray} &(67)_{10} \Rightarrow & 67 & |\underline{3} & & & \\\\ & & \fbox{1} & 22 & |\underline{3} & & \\\\ & & & \fbox{1} & 7 & |\underline{3} \\\\ & & & & \fbox{1} & \fbox{2} \end{eqnarray}$$
So
$$(3D)_{18}=(2111)_3$$
Solution:
$$(122)_3$$
$$(10022)_3$$
$$(11111)_3$$
$$(2111)_3$$