Convert from binary to quinary How to convert a number from binary to quinary system without using decimal system ? It is possible ?
I want to write a program who does it.
 A: In scanning a binary number (positive integer) from left to right, a $0$ bit doubles the previous value and a $1$ bit doubles the previous value and adds $1$.  
So in left to right scanning, say $1011_2$, values are one ($1$), two ($10$), five ($101$), eleven ($1011)$.  The value of this binary numeral is eleven.  This idea of doubling or doubling and adding one can be done in any base.
For instance $10111_2$; converting to base $5$ (quinary) would go $1, 2, 10, 21, 43$.  Answer $43_5$. 
Or $1000101_2$ to base $5$ would go $1, 2, 4, 13, 32, 114, 234$  Answer: $234_5$.
A: See Donald E. Knuth: The Art of Computer Programming,  Volume 2: Seminumerical Algorithms, third edition , chapter 4.4: Radix Conversion.
Knuth discusses four methods of conversion from base $B$ to $b$ using either $B$ or $b$ arithmetic.


*

*Method 1a: Division by $B$ using radix-$b$ arithmetic

*Method 1b: Multiplication by $b$ using radix-$B$ arithmetic

*Method 2a: Multiplication by $B$ using radix-$b$ arithmetic

*Method 2b: Division by $b$ using radix-$B$ arithmetic
Here neither $b$ nor $B$ must be $10$.
Note: You have probably access to arithmetic routines using base $b = 2$ (so you could avoid $b = 10$ as work arithmetic) and then radix convert to $B = 5$.
A: This question has already been treated in Computer Science Stackexchange.
See here: https://cs.stackexchange.com/questions/10318/the-math-behind-converting-from-any-base-to-any-base-without-going-through-base
