if y = 23$_{five}$ + 101$_{three}$ find y leaving your answer in base two

To solve this problem, we need to convert each number from its respective base into base ten, add them together, and then convert the result into base two.

Step 1: Convert $23_{five}$ to base ten

To convert from base five to base ten, we multiply each digit by 5 raised to the power of its position (starting from 0 on the right): $23_{five} = (2 \times 5^1) + (3 \times 5^0)$ $23_{five} = (2 \times 5) + (3 \times 1)$ $23_{five} = 10 + 3 = 13_{ten}$

Step 2: Convert $101_{three}$ to base ten

Similarly, for base three, we multiply each digit by 3 raised to the power of its position: $101_{three} = (1 \times 3^2) + (0 \times 3^1) + (1 \times 3^0)$ $101_{three} = (1 \times 9) + (0 \times 3) + (1 \times 1)$ $101_{three} = 9 + 0 + 1 = 10_{ten}$

Step 3: Add the base ten values

$y = 13_{ten} + 10_{ten} = 23_{ten}$

Step 4: Convert $23_{ten}$ to base two

To convert to base two, we perform repeated division by 2 and record the remainders:

• $23 \div 2 = 11$ remainder 1
• $11 \div 2 = 5$ remainder 1
• $5 \div 2 = 2$ remainder 1
• $2 \div 2 = 1$ remainder 0
• $1 \div 2 = 0$ remainder 1

Reading the remainders from bottom to top, we get 10111.

Conclusion

Comparing our result to the given options, we find that $10111_2$ matches option B.

Exam Tip: When converting from base ten to any other base, always remember to read your remainders from the last one calculated to the first one (bottom to top). A common mistake is reading them top to bottom, which will give you the wrong answer!