The product of the numbers 5623 and 6 in the scale of eight is

Options

A. 41672
B. 33738
C. 42562
D. 45262

Answer

To find the product of the numbers 5623 and 6 in the scale of eight (octal), we will follow these steps:

### Step 1: Understand the Octal System
The octal number system is a base-8 number system that uses digits from 0 to 7. In this system, the number 5623 is already in octal format.

### Step 2: Multiply Each Digit by 6
We will multiply each digit of the number 5623 by 6, starting from the rightmost digit (the least significant digit).

1. **Multiply the rightmost digit (3) by 6:**
\[ 6 \times 3 = 18 \]
We need to convert 18 into octal.
- \( 18 \div 8 = 2 \) remainder \( 2 \) (so we write down 2 and carry over 2).

2. **Multiply the next digit (2) by 6 and add the carry (2):**
\[ 6 \times 2 + 2 = 12 + 2 = 14 \]
Convert 14 into octal.
- \( 14 \div 8 = 1 \) remainder \( 6 \) (so we write down 6 and carry over 1).

3. **Multiply the next digit (6) by 6 and add the carry (1):**
\[ 6 \times 6 + 1 = 36 + 1 = 37 \]
Convert 37 into octal.
- \( 37 \div 8 = 4 \) remainder \( 5 \) (so we write down 5 and carry over 4).

4. **Multiply the leftmost digit (5) by 6 and add the carry (4):**
\[ 6 \times 5 + 4 = 30 + 4 = 34 \]
Convert 34 into octal.
- \( 34 \div 8 = 4 \) remainder \( 2 \) (so we write down 2 and carry over 4).

### Step 3: Compile the Result
Now we compile the results from each step, starting from the last carry (if any):
- From the last multiplication, we have a carry of 4.
- The results from the multiplications are:
- 4 (carry)
- 2
- 5
- 6
- 2

Thus, the final product in octal is:
\[ 42562_8 \]

### Final Answer
The product of the numbers 5623 and 6 in the scale of eight is \( 42562_8 \).

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