Changing a number between arbitrary bases

Let $x$ be a number. Then if $b$ is any base, $x \% b$ ($x$ mod $b$) is the last digit of $x$'s base-$b$ representation. Now integer-divide $x$ by $b$ to amputate the last digit.

Repeat and this procedure yields the digits of $x$ from least significant to most. It begins "little end first."

EDIT: Here is an example to make things clear.

Let $x = 45$ and $b = 3$.

x   x mod 3
45    0
15    0                (integer divide x by 3) 
 5    2
 1    1

We see that $45 = 1200_3$. Read up the last column to get the base-3 expansion you seek. Let us check.

$$1\cdot 3^3 + 2\cdot 3^2 + 0 + 0 = 27 + 18 = 45.$$

I hope this helps you.

You can perform base conversion directly by representing radix notation in horner (nested) form. Let's work a simply example. We convert $\:1213_{\:6}\:$ from radix $6$ to radix $8$

$$ 1{\color{red}2}{\color{blue}1}{\color{orange}3}_{\:6}\ =\ ((1\cdot 6+{\color{red}2})\:6+{\color{blue}1})\:6 + {\color{orange}3}$$

Now perform the computation inside-out in radix $8$:

$$ 1\cdot 6+ {\color{red}2} = 10)\: 6 = 60) + {\color{blue}1}) = 61)\: 6 = 446) + {\color{orange}3} = 451$$

Hence $\:1213_{\:\!6} = 451_{8}$