Convert a number from Base B1 to Base B2 without using any intermediate base

Question

Is there a way Convert a number from Base B1 to Base B2 without using any intermediate base.

Ex:

214 from base 5 to base 16 without converting it first to decima

Answer 1

To convert 214_base5 to base 16 without an intermediate base, you "just" have to know how to calculate directly in base 5.

First, you need a table of what the base 16 digits are in base 5 (you need a similar table when converting base 10 to base 16, it's just that that one is easier to keep in your head!). This table is easy to create - just start at 0 and increment each base 5 row until you reach f in base 16.

base 16 | base 5
--------+--------
      0 |  0
      1 |  1
      2 |  2
      3 |  3
      4 |  4
      5 | 10
      6 | 11
      7 | 12
      8 | 13
      9 | 14
      a | 20
      b | 21
      c | 22
      d | 23
      e | 24
      f | 30

Now you just need to repeatedly divide by 16 (which is 31_base5). We now recall our primary school days, and use long division (if this seems hard, it's because no-one made you learn your times-tables in base 5!):

Step 1:

   ______
31 ) 214

Step 2:

       3 
   ______
31 ) 214 -
     143  

Step 3:

       3 
   _____
31 ) 214 -
     143  
    ----
      21

So the result of 214_base5 divided by 31_base5 is 3_base5 remainder 21_base5.

This means that the least significant digit in base16 is 21_base5, which you can find in the table is b_base16. The result of the division is 3_base5 - if this was greater than 30_base5 then we would divide again - but it's not, so this means the most significant digit is (using the table again) 3_base16.

So the answer is 214_base5 = 3b_base16.