How do I convert a Base 3 number into a Base 5 number?

Question

How do I go about converting a Base 3 number into a Base 5 number, showing the steps needed to do it? Is it possible to do in one step? If you can explain, Many thanks, but thank you anyway if you can't.

Answer 1

It is a thought provoking question... Your question I read as " whether by any method can we convert a number with base 3 directly to base 5 with out intermediate conversion to base 10 ?.

By doing a little research, I found that it is possible

let 210 be a base 3 number ( equivalent base 10 being 21)

then for converting to base 5

Initially find the characteristic digits of base 5 ( the name that I have given for this purpose)
Corresponding to the highest digit 2 is 2
then second digit = second digit in base 3 minus 4 times first digit in base 3= 1-2*4=-7
The third digit = Third in base 3-2 times second plus 4 times highest = 0-2*1+4*2= 6

Then required 5 base # = 2 -7 6

bring to Base 5 by proper distribution-

6 as 11 etc it will be 041 to base 5 ( 41 is 21 to base 10 hence answer is verified)

Take another example 122 ( equivalent to 17 to base 10)

Then Characteristic digits in 5 base will

1 -2 2

which will proper rearranging will give 032 ( Equivalent to 17 to base 10)

Note : The rule stated above is only for three digit case but I have now general formula.. Which I am unable to write lack of time & space

Answer 2

rhsaunders is right: since we already think decimal, convert to decimal first. say you had 2012 base 3.

2012[3] = 2(3)^3 + 0(3)² + 1(3) + 2 = 54+3+2 = 59

59 / 5^3 = 0 r. 59
59 / 5² = 2 r. 9
9 / 5 = 1 r. 4
4 / 1 = 4 r. 0
so 59 = 214[5]

Answer 3

It is possible to do it in one step, but I am much more comfortable converting first to decimal -- which makes for two steps. Convert to decimal by adding appropriate multiples of 3^0, 3^1, 3^2, etc., and convert to quinary by subtracting multiples of 5*0, 5^1, etc but in reverse order -- it's basically a long division problem.