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

To solve this problem, we need to convert both numbers from their respective bases (base 5 and base 3) into base 10, add them together, and then convert the result into base 2.

Step 1: Convert $23_{five}$ to base 10

To convert from base $n$ to base 10, we multiply each digit by the base 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 10

$101_{three} = (1 \times 3^2) + (0 \times 3^1) + (1 \times 3^0)$ $101_{three} = (1 \times 9) + (0 \times 3) + (1 \times 1)$ $101_{three} = 9 + 0 + 1 = 10_{ten}$

Step 3: Add the two values in base 10

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

Step 4: Convert $23_{ten}$ to base 2

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 options provided, we see that $10111_2$ matches option B.

Exam Tip: When converting from base 10 to any other base, always remember to read your remainders from the bottom to the top (the last remainder is the Most Significant Digit). Also, ensure you don't confuse the powers of the base—always start from $0$ on the far right digit.