Number Bases and Conversions

21 questions

What is the result of the subtraction $423_6 - 145_6$ expressed in base 6?

234_6

232_6

344_6

322_6

Show answer & explanation

Answer: $234_6$

234_6

Subtracting

145_6

from

423_6

column by column requires borrowing from higher place values. In the units column, borrowing 1 (worth 6) gives

3 + 6 - 5 = 4

. In the sixes column, borrowing 1 leaves 1 six, and

1 + 6 - 4 = 3

. In the thirty-sixes column,

3 - 1 = 2

. Thus, the correct result is

234_6

Convert both base 6 numbers into base 10 (decimal).

423_6 = 4 \times 6^2 + 2 \times 6^1 + 3 \times 6^0 = 144 + 12 + 3 = 159_{10}

, and

145_6 = 1 \times 6^2 + 4 \times 6^1 + 5 \times 6^0 = 36 + 24 + 5 = 65_{10}

Converting both numbers to decimal provides a familiar base for subtraction.

Subtract the decimal numbers.

159_{10} - 65_{10} = 94_{10}

Perform standard decimal subtraction.

Convert the decimal result

94_{10}

back to base 6.

94 \div 6 = 15

4

;

15 \div 6 = 2

3

;

2 \div 6 = 0

2

. Arranging the remainders from bottom to top gives

234_6

Successive division by 6 extracts the base 6 digits.

Non-decimal base subtraction and conversions

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