To convert the binary number \(11011_2\) to base 5, we can first convert it to decimal (base 10), and then convert the decimal number to base 5.
Step 1: Convert from binary (base 2) to decimal (base 10)
The binary number \(11011_2\) can be converted to decimal by calculating the sum of the powers of 2, where each bit represents a power of 2:
\[ 11011_2 = 1 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0 \]
Calculating this step by step:
\[ = 1 \cdot 16 + 1 \cdot 8 + 0 \cdot 4 + 1 \cdot 2 + 1 \cdot 1 \] \[ = 16 + 8 + 0 + 2 + 1 \] \[ = 27 \]
So, \(11011_2 = 27_{10}\).
Step 2: Convert from decimal (base 10) to base 5
Next, we convert \(27_{10}\) to base 5 by repeatedly dividing by 5 and recording the remainders:
- \(27 \div 5 = 5\) with a remainder of \(2\)
- \(5 \div 5 = 1\) with a remainder of \(0\)
- \(1 \div 5 = 0\) with a remainder of \(1\)
Now, we collect the remainders from the last division to the first:
Reading from bottom to top, the remainders are \(1, 0, 2\), which gives us \(102_5\).
Final Result
Thus, the binary number \(11011_2\) converts to \(102_5\) in base 5.
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