Step-by-step conversion of decimal 136 to octal 210 using repeated division by 8.

The task is to convert the decimal number \( 136_{10} \) to its octal equivalent. The provided image illustrates the step-by-step process of converting a decimal number to an octal number using repeated division by 8 and collecting the remainders. Below, I will explain the solution in detail.

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Step-by-Step Solution:

1. Start with the decimal number:
\[
136_{10}
\]

2. Divide the number by 8 and record the quotient and remainder:
- Divide \( 136 \) by \( 8 \):
\[
136 \div 8 = 17 \quad \text{(quotient)} \quad \text{and} \quad 0 \quad \text{(remainder)}
\]
- Quotient: \( 17 \)
- Remainder: \( 0 \)

3. Repeat the process with the quotient from the previous step:
- Divide \( 17 \) by \( 8 \):
\[
17 \div 8 = 2 \quad \text{(quotient)} \quad \text{and} \quad 1 \quad \text{(remainder)}
\]
- Quotient: \( 2 \)
- Remainder: \( 1 \)

4. Repeat the process again:
- Divide \( 2 \) by \( 8 \):
\[
2 \div 8 = 0 \quad \text{(quotient)} \quad \text{and} \quad 2 \quad \text{(remainder)}
\]
- Quotient: \( 0 \)
- Remainder: \( 2 \)

5. Stop when the quotient becomes zero:
- At this point, the quotient is \( 0 \), so we stop the division process.

6. Collect the remainders in reverse order (from bottom to top):
- The remainders obtained are: \( 2, 1, 0 \).
- Reading these remainders from bottom to top gives the octal number:
\[
210
\]

7. Write the final answer:
- The decimal number \( 136_{10} \) is equivalent to the octal number \( 210_8 \).

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Explanation of the Process:

- Why divide by 8?
Octal is a base-8 number system, so dividing by 8 helps extract the digits of the octal number one by one.

- Why collect remainders in reverse order?
The remainders represent the digits of the octal number, starting from the least significant digit (rightmost) to the most significant digit (leftmost). Hence, they must be read in reverse order.

- When to stop?
Division continues until the quotient becomes zero. At this point, all digits of the octal number have been extracted.

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Final Answer:

\[
\boxed{(136)_{10} = (210)_{8}}
\]