Decimal — Binary — Octal — Hexadecimal

Question

ScanSolve AI Tutor · Accepted Answer

Here is the completed conversion table:

Decimal	Binary	Octal	Hexadecimal
33	100001	41	21
117	1110101	165	75
451	111000011	703	1C3
431	110101111	657	1AF

Here are the step-by-step conversions for each row:

Row 1: Given Decimal 33

Decimal to Binary:

Divide 33 by 2 repeatedly and record the remainders.

$$
33 \div 2 = 16 \text{ R } 1
$$
$$
16 \div 2 = 8 \text{ R } 0
$$
$$
8 \div 2 = 4 \text{ R } 0
$$
$$
4 \div 2 = 2 \text{ R } 0
$$
$$
2 \div 2 = 1 \text{ R } 0
$$
$$
1 \div 2 = 0 \text{ R } 1
$$

Read the remainders from bottom to top.
Binary: $\boxed{\text{100001}}$
- Decimal to Octal:
Divide 33 by 8 repeatedly and record the remainders.
$$ 33 \div 8 = 4 \text{ R } 1 $$ $$ 4 \div 8 = 0 \text{ R } 4 $$
Read the remainders from bottom to top.
Octal: $\boxed{\text{41}}$
- Decimal to Hexadecimal:
Divide 33 by 16 repeatedly and record the remainders.
$$ 33 \div 16 = 2 \text{ R } 1 $$ $$ 2 \div 16 = 0 \text{ R } 2 $$
Read the remainders from bottom to top.
Hexadecimal: $\boxed{\text{21}}$
Row 2: Given Binary 1110101
- Binary to Decimal:

Assign powers of 2 to each binary digit from right to left, starting with $2^0$ .

$$
1 \cdot 2^6 + 1 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0
$$

Calculate the sum.

$$
64 + 32 + 16 + 0 + 4 + 0 + 1 = 117
$$
Decimal: $\boxed{\text{117}}$

Binary to Octal:

Group the binary digits into sets of three from right to left.
Add leading zeros if necessary. $1 \ 110 \ 101$
Convert each group of three binary digits to its octal equivalent.
$$ 001_2 = 1_8 $$ $$ 110_2 = 6_8 $$ $$ 101_2 = 5_8 $$ Octal: $\boxed{\text{165}}$
- Binary to Hexadecimal:
Group the binary digits into sets of four from right to left.
Add leading zeros if necessary. $0111 \ 0101$
Convert each group of four binary digits to its hexadecimal equivalent.
$$ 0111_2 = 7_{16} $$ $$ 0101_2 = 5_{16} $$ Hexadecimal: $\boxed{\text{75}}$
Row 3: Given Octal 703
- Octal to Decimal:
Assign powers of 8 to each octal digit from right to left, starting with $8^0$ .
$$ 7 \cdot 8^2 + 0 \cdot 8^1 + 3 \cdot 8^0 $$

Calculate the sum.

$$
7 \cdot 64 + 0 \cdot 8 + 3 \cdot 1 = 448 + 0 + 3 = 451
$$
Decimal: $\boxed{\text{451}}$

Octal to Binary:

Convert each octal digit to its 3-bit binary equivalent.
$$ 7_8 = 111_2 $$ $$ 0_8 = 000_2 $$ $$ 3_8 = 011_2 $$
Combine the binary groups.
Binary: $\boxed{\text{111000011}}$
- Octal to Hexadecimal:
Convert Octal to Binary (as above).
$$ 703_8 = 111000011_2 $$
Group the binary digits into sets of four from right to left.
Add leading zeros if necessary. $0001 \ 1100 \ 0011$
Convert each group of four binary digits to its hexadecimal equivalent.
$$ 0001_2 = 1_{16} $$ $$ 1100_2 = C_{16} $$ $$ 0011_2 = 3_{16} $$ Hexadecimal: $\boxed{\text{1C3}}$
Row 4: Given Hexadecimal 1AF
- Hexadecimal to Decimal:
Assign powers of 16 to each hexadecimal digit from right to left, starting with $16^0$ .
(Note: A=10, F=15) $1 \cdot 16^2 + A \cdot 16^1 + F \cdot 16^0$ $1 \cdot 16^2 + 10 \cdot 16^1 + 15 \cdot 16^0$

Calculate the sum.

$$
1 \cdot 256 + 10 \cdot 16 + 15 \cdot 1 = 256 + 160 + 15 = 431
$$
Decimal: $\boxed{\text{431}}$

Hexadecimal to Binary:

Convert each hexadecimal digit to its 4-bit binary equivalent.
$$ 1_{16} = 0001_2 $$ $$ A_{16} = 1010_2 $$ $$ F_{16} = 1111_2 $$
Combine the binary groups.
Leading zeros can be omitted.
Binary: $\boxed{\text{110101111}}$
- Hexadecimal to Octal:
Convert Hexadecimal to Binary (as above).
$$ 1AF_{16} = 000110101111_2 $$
Group the binary digits into sets of three from right to left.
Add leading zeros if necessary. $000 \ 110 \ 101 \ 111$

Convert each group of three binary digits to its octal equivalent.

$$
000_2 = 0_8
$$
$$
110_2 = 6_8
$$
$$
101_2 = 5_8
$$
$$
111_2 = 7_8
$$
Octal: $\boxed{\text{657}}$

| Decimal | Binary | Octal | Hexadecimal |

Step-by-step solution

Divide 33 by 2 repeatedly and record the remainders.

Read the remainders from bottom to top.

Divide 33 by 8 repeatedly and record the remainders.

Read the remainders from bottom to top.

Divide 33 by 16 repeatedly and record the remainders.

Read the remainders from bottom to top.

Assign powers of 2 to each binary digit from right to left, starting with $2^0$ .

Calculate the sum.

Group the binary digits into sets of three from right to left.

Convert each group of three binary digits to its octal equivalent.

Group the binary digits into sets of four from right to left.

Convert each group of four binary digits to its hexadecimal equivalent.

Assign powers of 8 to each octal digit from right to left, starting with $8^0$ .

Calculate the sum.

Convert each octal digit to its 3-bit binary equivalent.

Combine the binary groups.

Convert Octal to Binary (as above).

Group the binary digits into sets of four from right to left.

Convert each group of four binary digits to its hexadecimal equivalent.

Assign powers of 16 to each hexadecimal digit from right to left, starting with $16^0$ .

Calculate the sum.

Convert each hexadecimal digit to its 4-bit binary equivalent.

Combine the binary groups.

Convert Hexadecimal to Binary (as above).

Group the binary digits into sets of three from right to left.

Convert each group of three binary digits to its octal equivalent.

Still stuck on this one?

More Computer Science Questions

| Decimal | Binary | Octal | Hexadecimal |

Step-by-step solution

Divide 33 by 2 repeatedly and record the remainders.

Read the remainders from bottom to top.

Divide 33 by 8 repeatedly and record the remainders.

Read the remainders from bottom to top.

Divide 33 by 16 repeatedly and record the remainders.

Read the remainders from bottom to top.

Assign powers of 2 to each binary digit from right to left, starting with 202^020.

Calculate the sum.

Group the binary digits into sets of three from right to left.

Convert each group of three binary digits to its octal equivalent.

Group the binary digits into sets of four from right to left.

Convert each group of four binary digits to its hexadecimal equivalent.

Assign powers of 8 to each octal digit from right to left, starting with 808^080.

Calculate the sum.

Convert each octal digit to its 3-bit binary equivalent.

Combine the binary groups.

Convert Octal to Binary (as above).

Group the binary digits into sets of four from right to left.

Convert each group of four binary digits to its hexadecimal equivalent.

Assign powers of 16 to each hexadecimal digit from right to left, starting with 16016^0160.

Calculate the sum.

Convert each hexadecimal digit to its 4-bit binary equivalent.

Combine the binary groups.

Convert Hexadecimal to Binary (as above).

Group the binary digits into sets of three from right to left.

Convert each group of three binary digits to its octal equivalent.

Still stuck on this one?

More Computer Science Questions

Assign powers of 2 to each binary digit from right to left, starting with $2^0$ .

Assign powers of 8 to each octal digit from right to left, starting with $8^0$ .

Assign powers of 16 to each hexadecimal digit from right to left, starting with $16^0$ .