Positional Number Systems

Question

I would ask you help for solution. The problem is: Convert number 3424 base 5 to base 6 in Positional Number Systems Thank you in Advance!

Answer 1

Here is one way of beginning what I think you are asking.

$$3424_5=3\times 5^3+4\times 5^2+2\times 5+4=3\times (6-1)^3+4\times (6-1)^2+2\times (6-1)+4$$

Then you can gather the powers of $6$ together, and most of the work is done. Whether this is easier or more efficient than other methods I'm not sure. It works well here because the bases are one apart.

You may know a method of successive division to convert base $10$ to base $2$, which would work generally. Its practical efficiency depends on knowing how to divide by $2$ in base $10$. Here the same approach would work, but you would be dividing by $6=11_5$ in base $5$. The computation above in fact does this for you, but organised in a different way.

One way of addressing such questions is to go via a familiar base in which arithmetic is easy (base $10$ would be favourite, but some people know things like bases $2, 8, 16$ well enough). That requires two calculations. Another would be to try to go via a special common base (here $30$ might be tried, as divisible by both $5$ and $6$) in an attempt to do two easy calculations.

I am trying to be helpful here, but you do need to clarify your question, and give some indication of the level of your mathematical background - you may understand what I have written - that would be good, or you may wonder what it is all about. You haven't, at the moment, told me enough about yourself for me to know how best to help you.