To solve the problem of dividing \(2434_6\) by \(42_6\), we first need to convert both numbers from base 6 to base 10 (decimal), perform the division in base 10, and then convert the result back to base 6. ### Step 1: Convert \(2434_6\) to Decimal To convert a number from base 6 to decimal, we multiply each digit by \(6\) raised to the power of its position (counting from right to left, starting at 0). For \(2434_6\): - The rightmost digit (4) is in the \(6^0\) place. - The next digit (3) is in the \(6^1\) place. - The next digit (4) is in the \(6^2\) place. - The leftmost digit (2) is in the \(6^3\) place. Calculating each term: - \(2 \times 6^3 = 2 \times 216 = 432\) - \(4 \times 6^2 = 4 \times 36 = 144\) - \(3 \times 6^1 = 3 \times 6 = 18\) - \(4 \times 6^0 = 4 \times 1 = 4\) Now, we sum these values: \[ 432 + 144 + 18 + 4 = 598 \] So, \(2434_6 = 598_{10}\). ### Step 2: Convert \(42_6\) to Decimal Now, we convert \(42_6\) to decimal using the same method: - The rightmost digit (2) is in the \(6^0\) place. - The leftmost digit (4) is in the \(6^1\) place. Calculating each term: - \(4 \times 6^1 = 4 \times 6 = 24\) - \(2 \times 6^0 = 2 \times 1 = 2\) Now, we sum these values: \[ 24 + 2 = 26 \] So, \(42_6 = 26_{10}\). ### Step 3: Perform the Division in Decimal Now we divide the two decimal numbers: \[ 598 \div 26 \] Calculating this gives: \[ 598 \div 26 = 23 \] This means that \(2434_6\) divided by \(42_6\) equals \(23\) in decimal. ### Step 4: Convert the Result Back to Base 6 Next, we need to convert \(23_{10}\) back to base 6. We do this by repeatedly dividing by 6 and keeping track of the remainders. 1. \(23 \div 6 = 3\) remainder \(5\) 2. \(3 \div 6 = 0\) remainder \(3\) Reading the remainders from bottom to top, we find that \(23_{10} = 35_6\). ### Step 5: Check the Options Now, we need to check the options provided: - A. 236 - B. 356 - C. 526 - D. 556 The correct answer, \(35_6\), is not listed among the options. However, if we consider the possibility of a typographical error in the options, we can see that option B, \(356\), is the closest match to our result \(35_6\) when considering the base representation. ### Conclusion The correct answer to the division of \(2434_6\) by \(42_6\) is \(35_6\), which corresponds to option B, \(356\), if we assume a typographical error. ### Summary - Convert numbers from base 6 to decimal for easier calculations. - Perform the division in decimal. - Convert the result back to base 6. - Check the options carefully for potential typographical errors. This thorough approach ensures that you understand each step of the process and can apply it to similar problems in the future.