Question

A particular number is written as $132$ in radix-$4$ representation. The same number in radix-$5$ representation is ________

Hint:

To convert between different bases, always first go through base 10 (decimal) unless a direct pattern is obvious. Divide by the target base and collect remainders bottom-up.

Answer 1

Step 1: Convert $132_{4$ to decimal.}
\[ 132_{4} = 1 4^2 + 3 4^1 + 2 4^0 \] \[ = 1 16 + 3 4 + 2 1 \] \[ = 16 + 12 + 2 = 30_{10}. \] Step 2: Convert decimal $30_{10$ to base-5.}
Divide by 5 repeatedly: \[ 30 \div 5 = 6 \text{remainder } 0 \] \[ 6 \div 5 = 1 \text{remainder } 1 \] \[ 1 \div 5 = 0 \text{remainder } 1 \] Reading remainders from bottom to top: \[ 30_{10} = 110_{5}. \] \[ \boxed{132_{4} = 110_{5}} \]