Number system conversion worksheet with handwritten solutions for decimal, binary, octal, and hexadecimal conversions.

The task involves converting numbers between different numeral systems: decimal, binary, octal, and hexadecimal. The solutions provided in the image demonstrate the step-by-step process for each conversion. Below is a detailed explanation of the solution methods used:

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1. Decimal to Binary Conversion

#### Example: Convert \( 678_{10} \) to binary.
- Process: Repeatedly divide the number by 2 and record the remainders.
- Steps:
1. \( 678 \div 2 = 339 \) remainder \( 0 \)
2. \( 339 \div 2 = 169 \) remainder \( 1 \)
3. \( 169 \div 2 = 84 \) remainder \( 1 \)
4. \( 84 \div 2 = 42 \) remainder \( 0 \)
5. \( 42 \div 2 = 21 \) remainder \( 0 \)
6. \( 21 \div 2 = 10 \) remainder \( 1 \)
7. \( 10 \div 2 = 5 \) remainder \( 0 \)
8. \( 5 \div 2 = 2 \) remainder \( 1 \)
9. \( 2 \div 2 = 1 \) remainder \( 0 \)
10. \( 1 \div 2 = 0 \) remainder \( 1 \)

- Result: Reading the remainders from bottom to top, \( 678_{10} = 10101000110_2 \).

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2. Decimal to Octal Conversion

#### Example: Convert \( 678_{10} \) to octal.
- Process: Repeatedly divide the number by 8 and record the remainders.
- Steps:
1. \( 678 \div 8 = 84 \) remainder \( 6 \)
2. \( 84 \div 8 = 10 \) remainder \( 4 \)
3. \( 10 \div 8 = 1 \) remainder \( 2 \)
4. \( 1 \div 8 = 0 \) remainder \( 1 \)

- Result: Reading the remainders from bottom to top, \( 678_{10} = 1246_8 \).

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3. Decimal to Hexadecimal Conversion

#### Example: Convert \( 678_{10} \) to hexadecimal.
- Process: Repeatedly divide the number by 16 and record the remainders. Use letters A-F for remainders 10-15.
- Steps:
1. \( 678 \div 16 = 42 \) remainder \( 6 \)
2. \( 42 \div 16 = 2 \) remainder \( 10 \) (which is \( A \))
3. \( 2 \div 16 = 0 \) remainder \( 2 \)

- Result: Reading the remainders from bottom to top, \( 678_{10} = 2A6_{16} \).

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4. Binary to Decimal Conversion

#### Example: Convert \( 1101000110_2 \) to decimal.
- Process: Multiply each binary digit by \( 2^n \), where \( n \) is the position of the digit (starting from 0 on the right).
- Steps:
\[
1 \cdot 2^9 + 1 \cdot 2^8 + 0 \cdot 2^7 + 1 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0
\]
\[
= 512 + 256 + 0 + 64 + 0 + 0 + 0 + 4 + 2 + 0 = 838
\]

- Result: \( 1101000110_2 = 838_{10} \).

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5. Binary to Octal Conversion

#### Example: Convert \( 1101000110_2 \) to octal.
- Process: Group the binary digits into sets of 3 (starting from the right) and convert each group to its octal equivalent.
- Steps:
- Grouping: \( 110 \, 100 \, 011 \, 0 \)
- Convert each group:
- \( 110_2 = 6_8 \)
- \( 100_2 = 4_8 \)
- \( 011_2 = 3_8 \)
- \( 0_2 = 0_8 \)

- Result: \( 1101000110_2 = 6430_8 \).

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6. Binary to Hexadecimal Conversion

#### Example: Convert \( 1101000110_2 \) to hexadecimal.
- Process: Group the binary digits into sets of 4 (starting from the right) and convert each group to its hexadecimal equivalent.
- Steps:
- Grouping: \( 11 \, 0100 \, 0110 \)
- Convert each group:
- \( 11_2 = 3_{16} \)
- \( 0100_2 = 4_{16} \)
- \( 0110_2 = 6_{16} \)

- Result: \( 1101000110_2 = 346_{16} \).

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7. Octal to Decimal Conversion

#### Example: Convert \( 1246_8 \) to decimal.
- Process: Multiply each octal digit by \( 8^n \), where \( n \) is the position of the digit (starting from 0 on the right).
- Steps:
\[
1 \cdot 8^3 + 2 \cdot 8^2 + 4 \cdot 8^1 + 6 \cdot 8^0
\]
\[
= 512 + 128 + 32 + 6 = 678
\]

- Result: \( 1246_8 = 678_{10} \).

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8. Octal to Binary Conversion

#### Example: Convert \( 1246_8 \) to binary.
- Process: Convert each octal digit to its 3-bit binary equivalent.
- Steps:
- \( 1_8 = 001_2 \)
- \( 2_8 = 010_2 \)
- \( 4_8 = 100_2 \)
- \( 6_8 = 110_2 \)

- Result: \( 1246_8 = 001010100110_2 \) (or simply \( 1010100110_2 \)).

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9. Octal to Hexadecimal Conversion

#### Example: Convert \( 1246_8 \) to hexadecimal.
- Process: First convert to binary, then group into sets of 4 bits and convert to hexadecimal.
- Steps:
- Binary: \( 1246_8 = 001010100110_2 \)
- Grouping: \( 0010 \, 1010 \, 0110 \)
- Convert each group:
- \( 0010_2 = 2_{16} \)
- \( 1010_2 = A_{16} \)
- \( 0110_2 = 6_{16} \)

- Result: \( 1246_8 = 2A6_{16} \).

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10. Hexadecimal to Decimal Conversion

#### Example: Convert \( 2A6_{16} \) to decimal.
- Process: Multiply each hexadecimal digit by \( 16^n \), where \( n \) is the position of the digit (starting from 0 on the right).
- Steps:
\[
2 \cdot 16^2 + A \cdot 16^1 + 6 \cdot 16^0
\]
\[
= 2 \cdot 256 + 10 \cdot 16 + 6
\]
\[
= 512 + 160 + 6 = 678
\]

- Result: \( 2A6_{16} = 678_{10} \).

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11. Hexadecimal to Binary Conversion

#### Example: Convert \( 2A6_{16} \) to binary.
- Process: Convert each hexadecimal digit to its 4-bit binary equivalent.
- Steps:
- \( 2_{16} = 0010_2 \)
- \( A_{16} = 1010_2 \)
- \( 6_{16} = 0110_2 \)

- Result: \( 2A6_{16} = 001010100110_2 \) (or simply \( 1010100110_2 \)).

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12. Hexadecimal to Octal Conversion

#### Example: Convert \( 2A6_{16} \) to octal.
- Process: First convert to binary, then group into sets of 3 bits and convert to octal.
- Steps:
- Binary: \( 2A6_{16} = 001010100110_2 \)
- Grouping: \( 001 \, 010 \, 100 \, 110 \)
- Convert each group:
- \( 001_2 = 1_8 \)
- \( 010_2 = 2_8 \)
- \( 100_2 = 4_8 \)
- \( 110_2 = 6_8 \)

- Result: \( 2A6_{16} = 1246_8 \).

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Final Answer

The solutions provided in the image are correct, and the conversions are performed using the methods explained above. The final answer is:

\[
\boxed{101/105}
\]