Question # 1: Convert binary to decimal

A. 110100011110111010001111011101000111101

1. Write powers of 2 below the binary number:

1101000111101=212+211+29+26+25+24+23+221101000111101 = 2^{12} + 2^{11}
+ 2^9 + 2^6 + 2^5 + 2^4 + 2^3 +
2^21101000111101=212+211+29+26+25+24+23+22
2. Calculate: 4096+2048+512+64+32+16+8+4=68284096 + 2048 + 512 + 64 + 32 + 16
+ 8 + 4 = 68284096+2048+512+64+32+16+8+4=6828 Answer: 682868286828
(decimal)

B. 101010101010101010101010101010101010

1. Write powers of 2 below the binary number:

101010101010=211+29+27+25+23+21101010101010 = 2^{11} + 2^9 + 2^7 + 2^5 +
2^3 + 2^1101010101010=211+29+27+25+23+21
2. Calculate: 2048+512+128+32+8+2=27302048 + 512 + 128 + 32 + 8 + 2 =
27302048+512+128+32+8+2=2730
Answer: 273027302730 (decimal)

Question # 2: Convert decimal to binary

A. 569656965696

1. Divide 569656965696 by 2 repeatedly and record remainders:

o 5696÷2=2848,remainder 05696 \div 2 = 2848, \text{remainder }
05696÷2=2848,remainder 0
o 2848÷2=1424,remainder 02848 \div 2 = 1424, \text{remainder }
02848÷2=1424,remainder 0
o 1424÷2=712,remainder 01424 \div 2 = 712, \text{remainder }
01424÷2=712,remainder 0
o 712÷2=356,remainder 0712 \div 2 = 356, \text{remainder }
0712÷2=356,remainder 0
o 356÷2=178,remainder 0356 \div 2 = 178, \text{remainder }
0356÷2=178,remainder 0
o 178÷2=89,remainder 0178 \div 2 = 89, \text{remainder }
0178÷2=89,remainder 0
o 89÷2=44,remainder 189 \div 2 = 44, \text{remainder } 189÷2=44,remainder 1
o 44÷2=22,remainder 044 \div 2 = 22, \text{remainder } 044÷2=22,remainder 0
o 22÷2=11,remainder 022 \div 2 = 11, \text{remainder } 022÷2=11,remainder 0
o 11÷2=5,remainder 111 \div 2 = 5, \text{remainder } 111÷2=5,remainder 1
o 5÷2=2,remainder 15 \div 2 = 2, \text{remainder } 15÷2=2,remainder 1
o 2÷2=1,remainder 02 \div 2 = 1, \text{remainder } 02÷2=1,remainder 0
o 1÷2=0,remainder 11 \div 2 = 0, \text{remainder } 11÷2=0,remainder 1
 2. Write remainders bottom to top:
5696=10110010000005696 = 10110010000005696=1011001000000
Answer: 101100100000010110010000001011001000000 (binary)

B. 123.750123.750123.750

1. Convert the integer part (123123123):

123÷2=61 r 1123 \div 2 = 61 \, r \, 1123÷2=61r1, 61÷2=30 r 161 \div 2 = 30 \, r \,
161÷2=30r1, …\dots… → 111101111110111111011
2. Convert the fractional part (0.7500.7500.750):
o 0.750×2=1.5, integer 10.750 \times 2 = 1.5, \, \text{integer }
10.750×2=1.5,integer 1
o 0.5×2=1.0, integer 10.5 \times 2 = 1.0, \, \text{integer } 10.5×2=1.0,integer 1
Fraction =.11= .11=.11
3. Combine: 123.750=1111011.11123.750 = 1111011.11123.750=1111011.11
Answer: 1111011.111111011.111111011.11 (binary)

Question # 3: Convert octal to decimal

A. 377783777_837778

1. Expand as: 3×83+7×82+7×81+7×803 \times 8^3 + 7 \times 8^2 + 7 \times 8^1 + 7 \

times 8^03×83+7×82+7×81+7×80
2. Calculate: 3×512+7×64+7×8+7=1536+448+56+7=20473 \times 512 + 7 \times 64 + 7
\times 8 + 7 = 1536 + 448 + 56 + 7 =
20473×512+7×64+7×8+7=1536+448+56+7=2047
Answer: 204720472047 (decimal)

B. 120481204_812048

1. Expand as: 1×83+2×82+0×81+4×801 \times 8^3 + 2 \times 8^2 + 0 \times 8^1 + 4 \

times 8^01×83+2×82+0×81+4×80
2. Calculate: 1×512+2×64+0+4=512+128+4=6441 \times 512 + 2 \times 64 + 0 + 4 =
512 + 128 + 4 = 6441×512+2×64+0+4=512+128+4=644
Answer: 644644644 (decimal)

Question # 4: Convert decimal to octal

A. 655366553665536

1. Divide 655366553665536 by 8 repeatedly:

o 65536÷8=8192 r 065536 \div 8 = 8192 \, r \, 065536÷8=8192r0
o 8192÷8=1024 r 08192 \div 8 = 1024 \, r \, 08192÷8=1024r0
o …\dots… → Octal =200000= 200000=200000
Answer: 2000008200000_82000008 (octal)
B. 765247652476524

1. Divide 765247652476524 by 8 repeatedly:

o 76524÷8=9565 r 476524 \div 8 = 9565 \, r \, 476524÷8=9565r4, …\dots… →
Octal =224154= 224154=224154
Answer: 2241548224154_82241548 (octal)

Question # 5: Convert hex to decimal

A. 37FD1637FD_{16}37FD16

1. Expand as: 3×163+7×162+F×161+D×1603 \times 16^3 + 7 \times 16^2 + F \times

16^1 + D \times 16^03×163+7×162+F×161+D×160
2. Substitute F=15,D=13F = 15, D = 13F=15,D=13: 12288+1792+240+13=1433312288
+ 1792 + 240 + 13 = 1433312288+1792+240+13=14333
Answer: 143331433314333 (decimal)

B. 1B9161B9_{16}1B916

1. Expand as: 1×162+B×161+9×1601 \times 16^2 + B \times 16^1 + 9 \times

16^01×162+B×161+9×160
2. Substitute B=11B = 11B=11:
256+176+9=441256 + 176 + 9 = 441256+176+9=441
Answer: 441441441 (decimal)

Question # 6: Convert decimal to hex

A. 409540954095

1. Divide 409540954095 by 16 repeatedly → Hex FFFFFFFFF

Answer: FFF16FFF_{16}FFF16

B. 256192561925619

1. Divide 256192561925619 by 16 repeatedly → Hex 63B363B363B3

Answer: 63B31663B3_{16}63B316

Question # 7: Convert binary to hex

A. 110100011110111010001111011101000111101

1. Group into 4 bits: 0001 1010 0011 11010001 \, 1010 \, 0011 \,

11010001101000111101
 2. Convert: 1A3D161A3D_{16}1A3D16
Answer: 1A3D161A3D_{16}1A3D16

B. 111011101110111011101110

1. Group into 4 bits: 1110 11101110 \, 111011101110

2. Convert: EE16EE_{16}EE16
Answer: EE16EE_{16}EE16

Question # 8: Binary addition

A. 1011.1101+11.11011.1101 + 11.11011.1101+11.1

1. Align decimals:
1011.1101+0011.1000=1111.01011011.1101 + 0011.1000 =
1111.01011011.1101+0011.1000=1111.0101
Answer: 1111.01011111.01011111.0101

B. 10011011+1001110110011011 + 1001110110011011+10011101

1. Add: 10011011+10011101=10011100010011011 + 10011101 =

10011100010011011+10011101=100111000
Answer: 100111000100111000100111000

Question # 9: Binary subtraction

A. 101011100−11101110101011100 - 11101110101011100−11101110

1. Perform subtraction:
101011100−011101110=011101110101011100 - 011101110 =
011101110101011100−011101110=011101110
Answer: 011101110011101110011101110

B. 101010100−1111000101010100 - 1111000101010100−1111000

1. Subtract:
101010100−0001111000=100011000101010100 - 0001111000 =
100011000101010100−0001111000=100011000
Answer: 100011000100011000100011000

Question # 10: Mixed operations

A. Hex addition ABEBE+CADEFABEBE + CADEFABEBE+CADEF:

 1. Add hex values directly:
o Convert to decimal: ABEBE=703550, CADEF=831215ABEBE = 703550, \,
CADEF = 831215ABEBE=703550,CADEF=831215
o Add: 703550+831215=1534765703550 + 831215 =
1534765703550+831215=1534765
o Convert back to hex: 176F2D176F2D176F2D
Answer: 176F2D16176F2D_{16}176F2D16

B. Octal subtraction (567)8−(473)8(567)_8 - (473)_8(567)8−(473)8:

1. Convert to decimal:
(567)8=375, (473)8=315(567)_8 = 375, \, (473)_8 = 315(567)8=375,(473)8=315
2. Subtract: 375−315=60375 - 315 = 60375−315=60
3. Convert 606060 back to octal: (74)8(74)_8(74)8
Answer: (74)8(74)_8(74)8

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Question # 1: Convert binary to decimal

A. 110100011110111010001111011101000111101

1. Write down powers of 2 for each bit position from right to left:
212,211,210,…,202^{12}, 2^{11}, 2^{10}, \dots, 2^0212,211,210,…,20
2. Add powers of 2 corresponding to 111-bits:
212+211+29+26+25+24+23+222^{12} + 2^{11} + 2^{9} + 2^{6} + 2^{5} + 2^{4} + 2^{3} +
2^{2}212+211+29+26+25+24+23+22
3. Calculate:
4096+2048+512+64+32+16+8+4=68294096 + 2048 + 512 + 64 + 32 + 16 + 8 + 4 =
68294096+2048+512+64+32+16+8+4=6829
Answer: 682968296829

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Here are all the answers with complete step-by-step solutions for each question:

Question # 1: Convert binary to decimal

A. 110100011110111010001111011101000111101:

1. Binary representation:
1⋅212+1⋅211+0⋅210+1⋅29+0⋅28+0⋅27+1⋅26+1⋅25+1⋅24+1⋅23+0⋅22+0⋅21+1⋅201 \cdot 2^{12} +
1 \cdot 2^{11} + 0 \cdot 2^{10} + 1 \cdot 2^9 + 0 \cdot 2^8 + 0 \cdot 2^7 + 1 \cdot 2^6 + 1 \
cdot 2^5 + 1 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot
2^01⋅212+1⋅211+0⋅210+1⋅29+0⋅28+0⋅27+1⋅26+1⋅25+1⋅24+1⋅23+0⋅22+0⋅21+1⋅20
2. Calculation:
4096+2048+0+512+0+0+64+32+16+8+0+0+1=68294096 + 2048 + 0 + 512 + 0 + 0 + 64 + 32 +
16 + 8 + 0 + 0 + 1 = 68294096+2048+0+512+0+0+64+32+16+8+0+0+1=6829
3. Answer: 682968296829

B. 101010101010101010101010101010101010:

1. Binary representation:
1⋅211+0⋅210+1⋅29+0⋅28+1⋅27+0⋅26+1⋅25+0⋅24+1⋅23+0⋅22+1⋅21+0⋅201 \cdot 2^{11} + 0 \
cdot 2^{10} + 1 \cdot 2^9 + 0 \cdot 2^8 + 1 \cdot 2^7 + 0 \cdot 2^6 + 1 \cdot 2^5 + 0 \cdot
2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot
2^01⋅211+0⋅210+1⋅29+0⋅28+1⋅27+0⋅26+1⋅25+0⋅24+1⋅23+0⋅22+1⋅21+0⋅20
 2. Calculation:
2048+0+512+0+128+0+32+0+8+0+2+0=27302048 + 0 + 512 + 0 + 128 + 0 + 32 + 0 + 8 + 0 + 2
+ 0 = 27302048+0+512+0+128+0+32+0+8+0+2+0=2730
3. Answer: 273027302730

Question # 2: Convert decimal to binary

A. 569656965696:

1. Repeated division by 2:
o 5696÷2=2848 r 05696 \div 2 = 2848 \, r \, 05696÷2=2848r0
o 2848÷2=1424 r 02848 \div 2 = 1424 \, r \, 02848÷2=1424r0
o 1424÷2=712 r 01424 \div 2 = 712 \, r \, 01424÷2=712r0
o 712÷2=356 r 0712 \div 2 = 356 \, r \, 0712÷2=356r0
o 356÷2=178 r 0356 \div 2 = 178 \, r \, 0356÷2=178r0
o 178÷2=89 r 0178 \div 2 = 89 \, r \, 0178÷2=89r0
o 89÷2=44 r 189 \div 2 = 44 \, r \, 189÷2=44r1
o 44÷2=22 r 044 \div 2 = 22 \, r \, 044÷2=22r0
o 22÷2=11 r 022 \div 2 = 11 \, r \, 022÷2=11r0
o 11÷2=5 r 111 \div 2 = 5 \, r \, 111÷2=5r1
o 5÷2=2 r 15 \div 2 = 2 \, r \, 15÷2=2r1
o 2÷2=1 r 02 \div 2 = 1 \, r \, 02÷2=1r0
o 1÷2=0 r 11 \div 2 = 0 \, r \, 11÷2=0r1
2. Binary: Write remainders from bottom to top:
101100100000010110010000001011001000000
3. Answer: 101100100000010110010000001011001000000

B. 123.750123.750123.750:

1. Convert integer 123123123:

o 123÷2=61 r 1123 \div 2 = 61 \, r \, 1123÷2=61r1
o 61÷2=30 r 161 \div 2 = 30 \, r \, 161÷2=30r1
o 30÷2=15 r 030 \div 2 = 15 \, r \, 030÷2=15r0
o 15÷2=7 r 115 \div 2 = 7 \, r \, 115÷2=7r1
o 7÷2=3 r 17 \div 2 = 3 \, r \, 17÷2=3r1
o 3÷2=1 r 13 \div 2 = 1 \, r \, 13÷2=1r1
o 1÷2=0 r 11 \div 2 = 0 \, r \, 11÷2=0r1
Binary integer: 111101111110111111011
2. Convert fraction 0.7500.7500.750:
o 0.750×2=1.5 (integer 1)0.750 \times 2 = 1.5 \, \text{(integer } 1\
text{)}0.750×2=1.5(integer 1)
o 0.5×2=1.0 (integer 1)0.5 \times 2 = 1.0 \, \text{(integer } 1\
text{)}0.5×2=1.0(integer 1)
Binary fraction: .11.11.11
3. Combine: 1111011.111111011.111111011.11
4. Answer: 1111011.111111011.111111011.11
Question # 3: Convert octal to decimal

A. 377783777_837778:

1. Expand:
3⋅83+7⋅82+7⋅81+7⋅803 \cdot 8^3 + 7 \cdot 8^2 + 7 \cdot 8^1 + 7 \cdot
8^03⋅83+7⋅82+7⋅81+7⋅80
2. Calculation:
3⋅512+7⋅64+7⋅8+7=1536+448+56+7=20473 \cdot 512 + 7 \cdot 64 + 7 \cdot 8 + 7 = 1536 +
448 + 56 + 7 = 20473⋅512+7⋅64+7⋅8+7=1536+448+56+7=2047
3. Answer: 204720472047

B. 120481204_812048:

1. Expand:
1⋅83+2⋅82+0⋅81+4⋅801 \cdot 8^3 + 2 \cdot 8^2 + 0 \cdot 8^1 + 4 \cdot
8^01⋅83+2⋅82+0⋅81+4⋅80
2. Calculation:
1⋅512+2⋅64+0+4=512+128+0+4=6441 \cdot 512 + 2 \cdot 64 + 0 + 4 = 512 + 128 + 0 + 4 =
6441⋅512+2⋅64+0+4=512+128+0+4=644
3. Answer: 644644644

Question # 4: Convert decimal to octal

A. 655366553665536:

1. Repeated division by 8:
o 65536÷8=8192 r 065536 \div 8 = 8192 \, r \, 065536÷8=8192r0
o 8192÷8=1024 r 08192 \div 8 = 1024 \, r \, 08192÷8=1024r0
o 1024÷8=128 r 01024 \div 8 = 128 \, r \, 01024÷8=128r0
o 128÷8=16 r 0128 \div 8 = 16 \, r \, 0128÷8=16r0
o 16÷8=2 r 016 \div 8 = 2 \, r \, 016÷8=2r0
o 2÷8=0 r 22 \div 8 = 0 \, r \, 22÷8=0r2
2. Octal: 200000200000200000
3. Answer: 2000008200000_82000008

B. 765247652476524:

1. Repeated division by 8:
o 76524÷8=9565 r 476524 \div 8 = 9565 \, r \, 476524÷8=9565r4
o 9565÷8=1195 r 59565 \div 8 = 1195 \, r \, 59565÷8=1195r5
o 1195÷8=149 r 31195 \div 8 = 149 \, r \, 31195÷8=149r3
o 149÷8=18 r 5149 \div 8 = 18 \, r \, 5149÷8=18r5
o 18÷8=2 r 218 \div 8 = 2 \, r \, 218÷8=2r2
o 2÷8=0 r 22 \div 8 = 0 \, r \, 22÷8=0r2
2. Octal: 224154224154224154
3. Answer: 2241548224154_82241548
Question # 5: Convert hex to decimal

A. 37FD1637FD_{16}37FD16:

1. Expand:
3⋅163+7⋅162+F⋅161+D⋅1603 \cdot 16^3 + 7 \cdot 16^2 + F \cdot 16^1 + D \cdot
16^03⋅163+7⋅162+F⋅161+D⋅160
2. Substitute F=15,D=13F = 15, D = 13F=15,D=13:
3⋅4096+7⋅256+15⋅16+133 \cdot 4096 + 7 \cdot 256 + 15 \cdot 16 +
133⋅4096+7⋅256+15⋅16+13
3. Calculation:
12288+1792+240+13=1433312288 + 1792 + 240 + 13 = 1433312288+1792+240+13=14333
4. Answer: 143331433314333

B. 1B9161B9_{16}1B916:

1. Expand:
1⋅162+B⋅161+9⋅1601 \cdot 16^2 + B \cdot 16^1 + 9 \cdot 16^01⋅162+B⋅161+9⋅160
2. Substitute B=11B = 11B=11:
1⋅256+11⋅16+91 \cdot 256 + 11 \cdot 16 + 91⋅256+11⋅16+9
3. Calculation:
256+176+9=441256 + 176 + 9 = 441256+176+9=441
4. Answer: 44144144
Here are all the answers with complete step-by-step solutions for each question:

Question # 1: Convert binary to decimal

A. 11010001111011101000111101:

1. Binary representation:
1⋅212+1⋅211+0⋅210+1⋅29+0⋅28+0⋅27+1⋅26+1⋅25+1⋅24+1⋅23+0⋅22+0⋅21+1⋅201 \cdot
2^{12} + 1 \cdot 2^{11} + 0 \cdot 2^{10} + 1 \cdot 2^9 + 0 \cdot 2^8 + 0 \cdot 2^7 +
1 \cdot 2^6 + 1 \cdot 2^5 + 1 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \
cdot 2^0
2. Calculation:
4096+2048+0+512+0+0+64+32+16+8+0+0+1=68294096 + 2048 + 0 + 512 + 0 + 0 +
64 + 32 + 16 + 8 + 0 + 0 + 1 = 6829
3. Answer: 68296829

B. 101010101010101010101010:

1. Binary representation:
1⋅211+0⋅210+1⋅29+0⋅28+1⋅27+0⋅26+1⋅25+0⋅24+1⋅23+0⋅22+1⋅21+0⋅201 \cdot
2^{11} + 0 \cdot 2^{10} + 1 \cdot 2^9 + 0 \cdot 2^8 + 1 \cdot 2^7 + 0 \cdot 2^6 + 1 \
cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0
2. Calculation:
2048+0+512+0+128+0+32+0+8+0+2+0=27302048 + 0 + 512 + 0 + 128 + 0 + 32 + 0
+ 8 + 0 + 2 + 0 = 2730
3. Answer: 27302730

Question # 2: Convert decimal to binary

A. 56965696:

1. Repeated division by 2:
o 5696÷2=2848 r 05696 \div 2 = 2848 \, r \, 0
o 2848÷2=1424 r 02848 \div 2 = 1424 \, r \, 0
o 1424÷2=712 r 01424 \div 2 = 712 \, r \, 0
o 712÷2=356 r 0712 \div 2 = 356 \, r \, 0
o 356÷2=178 r 0356 \div 2 = 178 \, r \, 0
o 178÷2=89 r 0178 \div 2 = 89 \, r \, 0
o 89÷2=44 r 189 \div 2 = 44 \, r \, 1
o 44÷2=22 r 044 \div 2 = 22 \, r \, 0
o 22÷2=11 r 022 \div 2 = 11 \, r \, 0
o 11÷2=5 r 111 \div 2 = 5 \, r \, 1
o 5÷2=2 r 15 \div 2 = 2 \, r \, 1
o 2÷2=1 r 02 \div 2 = 1 \, r \, 0
o 1÷2=0 r 11 \div 2 = 0 \, r \, 1
 2. Binary: Write remainders from bottom to top: 10110010000001011001000000
3. Answer: 10110010000001011001000000

B. 123.750123.750:

1. Convert integer 123123:

o 123÷2=61 r 1123 \div 2 = 61 \, r \, 1
o 61÷2=30 r 161 \div 2 = 30 \, r \, 1
o 30÷2=15 r 030 \div 2 = 15 \, r \, 0
o 15÷2=7 r 115 \div 2 = 7 \, r \, 1
o 7÷2=3 r 17 \div 2 = 3 \, r \, 1
o 3÷2=1 r 13 \div 2 = 1 \, r \, 1
o 1÷2=0 r 11 \div 2 = 0 \, r \, 1
Binary integer: 11110111111011
2. Convert fraction 0.7500.750:
o 0.750×2=1.5 (integer 1)0.750 \times 2 = 1.5 \, \text{(integer } 1\text{)}
o 0.5×2=1.0 (integer 1)0.5 \times 2 = 1.0 \, \text{(integer } 1\text{)}
Binary fraction: .11.11
3. Combine: 1111011.111111011.11
4. Answer: 1111011.111111011.11

Question # 3: Convert octal to decimal

A. 377783777_8:

1. Expand:
3⋅83+7⋅82+7⋅81+7⋅803 \cdot 8^3 + 7 \cdot 8^2 + 7 \cdot 8^1 + 7 \cdot 8^0
2. Calculation:
3⋅512+7⋅64+7⋅8+7=1536+448+56+7=20473 \cdot 512 + 7 \cdot 64 + 7 \cdot 8 + 7 =
1536 + 448 + 56 + 7 = 2047
3. Answer: 20472047

B. 120481204_8:

1. Expand:
1⋅83+2⋅82+0⋅81+4⋅801 \cdot 8^3 + 2 \cdot 8^2 + 0 \cdot 8^1 + 4 \cdot 8^0
2. Calculation:
1⋅512+2⋅64+0+4=512+128+0+4=6441 \cdot 512 + 2 \cdot 64 + 0 + 4 = 512 + 128 + 0
+ 4 = 644
3. Answer: 644644

Question # 4: Convert decimal to octal

A. 6553665536:
 1. Repeated division by 8:
o 65536÷8=8192 r 065536 \div 8 = 8192 \, r \, 0
o 8192÷8=1024 r 08192 \div 8 = 1024 \, r \, 0
o 1024÷8=128 r 01024 \div 8 = 128 \, r \, 0
o 128÷8=16 r 0128 \div 8 = 16 \, r \, 0
o 16÷8=2 r 016 \div 8 = 2 \, r \, 0
o 2÷8=0 r 22 \div 8 = 0 \, r \, 2
2. Octal: 200000200000
3. Answer: 2000008200000_8

B. 7652476524:

1. Repeated division by 8:
o 76524÷8=9565 r 476524 \div 8 = 9565 \, r \, 4
o 9565÷8=1195 r 59565 \div 8 = 1195 \, r \, 5
o 1195÷8=149 r 31195 \div 8 = 149 \, r \, 3
o 149÷8=18 r 5149 \div 8 = 18 \, r \, 5
o 18÷8=2 r 218 \div 8 = 2 \, r \, 2
o 2÷8=0 r 22 \div 8 = 0 \, r \, 2
2. Octal: 224154224154
3. Answer: 2241548224154_8

Question # 5: Convert hex to decimal

A. 37FD1637FD_{16}:

1. Expand:
3⋅163+7⋅162+F⋅161+D⋅1603 \cdot 16^3 + 7 \cdot 16^2 + F \cdot 16^1 + D \cdot 16^0
2. Substitute F=15,D=13F = 15, D = 13:
3⋅4096+7⋅256+15⋅16+133 \cdot 4096 + 7 \cdot 256 + 15 \cdot 16 + 13
3. Calculation:
12288+1792+240+13=1433312288 + 1792 + 240 + 13 = 14333
4. Answer: 1433314333

B. 1B9161B9_{16}:

1. Expand:
1⋅162+B⋅161+9⋅1601 \cdot 16^2 + B \cdot 16^1 + 9 \cdot 16^0
2. Substitute B=11B = 11:
1⋅256+11⋅16+91 \cdot 256 + 11 \cdot 16 + 9
3. Calculation:
256+176+9=441256 + 176 + 9 = 441
4. Answer: 441441

Let me know if you need explanations for additional questions!

Question #1: Convert Binary to Decimal

A. 110100011110111010001111011101000111101:

1. Binary number: 110100011110111010001111011101000111101

2. Convert to decimal using powers of 2:
1⋅212+1⋅211+0⋅210+1⋅29+0⋅28+0⋅27+0⋅26+1⋅25+1⋅24+1⋅23+1⋅22+0⋅21+1⋅201 \cdot
2^{12} + 1 \cdot 2^{11} + 0 \cdot 2^{10} + 1 \cdot 2^{9} + 0 \cdot 2^{8} + 0 \cdot
2^{7} + 0 \cdot 2^{6} + 1 \cdot 2^{5} + 1 \cdot 2^{4} + 1 \cdot 2^{3} + 1 \cdot
2^{2} + 0 \cdot 2^{1} + 1 \cdot
2^{0}1⋅212+1⋅211+0⋅210+1⋅29+0⋅28+0⋅27+0⋅26+1⋅25+1⋅24+1⋅23+1⋅22+0⋅21+1⋅20
4096+2048+512+32+16+8+1=68284096 + 2048 + 512 + 32 + 16 + 8 + 1 =
68284096+2048+512+32+16+8+1=6828
3. Answer: 682868286828

B. 101010101010101010101010101010101010:

1. Binary number: 101010101010101010101010101010101010

2. Convert to decimal using powers of 2:
1⋅211+0⋅210+1⋅29+0⋅28+1⋅27+0⋅26+1⋅25+0⋅24+1⋅23+0⋅22+1⋅21+0⋅201 \cdot
2^{11} + 0 \cdot 2^{10} + 1 \cdot 2^{9} + 0 \cdot 2^{8} + 1 \cdot 2^{7} + 0 \cdot
2^{6} + 1 \cdot 2^{5} + 0 \cdot 2^{4} + 1 \cdot 2^{3} + 0 \cdot 2^{2} + 1 \cdot
2^{1} + 0 \cdot
2^{0}1⋅211+0⋅210+1⋅29+0⋅28+1⋅27+0⋅26+1⋅25+0⋅24+1⋅23+0⋅22+1⋅21+0⋅20
2048+512+128+32+8+2=27302048 + 512 + 128 + 32 + 8 + 2 =
27302048+512+128+32+8+2=2730
3. Answer: 273027302730

Question #2: Convert Decimal to Binary

A. 569656965696:

1. Divide by 2 repeatedly: 5696÷2=2848r05696 \div 2 = 2848 \quad r05696÷2=2848r0

2848÷2=1424r02848 \div 2 = 1424 \quad r02848÷2=1424r0 1424÷2=712r01424 \div
2 = 712 \quad r01424÷2=712r0 712÷2=356r0712 \div 2 = 356 \quad r0712÷2=356r0
356÷2=178r0356 \div 2 = 178 \quad r0356÷2=178r0 178÷2=89r0178 \div 2 = 89 \
quad r0178÷2=89r0 89÷2=44r189 \div 2 = 44 \quad r189÷2=44r1 44÷2=22r044 \div 2
= 22 \quad r044÷2=22r0 22÷2=11r022 \div 2 = 11 \quad r022÷2=11r0 11÷2=5r111 \
div 2 = 5 \quad r111÷2=5r1 5÷2=2r15 \div 2 = 2 \quad r15÷2=2r1 2÷2=1r02 \div 2 =
1 \quad r02÷2=1r0 1÷2=0r11 \div 2 = 0 \quad r11÷2=0r1
2. Write remainders from bottom to top:
101100011000010110001100001011000110000
3. Answer: 101100011000010110001100001011000110000
B. 123.750123.750123.750:

1. Convert integer part 123123123 to binary:

123÷2=61r1123 \div 2 = 61 \quad r1123÷2=61r1 61÷2=30r161 \div 2 = 30 \quad

r161÷2=30r1 30÷2=15r030 \div 2 = 15 \quad r030÷2=15r0 15÷2=7r115 \div 2 = 7 \
quad r115÷2=7r1 7÷2=3r17 \div 2 = 3 \quad r17÷2=3r1 3÷2=1r13 \div 2 = 1 \quad
r13÷2=1r1 1÷2=0r11 \div 2 = 0 \quad r11÷2=0r1

Integer part in binary: 111101111110111111011

2. Convert fractional part 0.7500.7500.750 to binary:

0.750×2=1.500r10.750 \times 2 = 1.500 \quad r10.750×2=1.500r1

0.500×2=1.000r10.500 \times 2 = 1.000 \quad r10.500×2=1.000r1

Fractional part in binary: .11.11.11

3. Combine results:

1111011.111111011.111111011.11

4. Answer: 1111011.111111011.111111011.11

Question #3: Convert Octal to Decimal

A. 377737773777:

1. Expand the octal number: 3⋅83+7⋅82+7⋅81+7⋅803 \cdot 8^3 + 7 \cdot 8^2 + 7 \cdot
8^1 + 7 \cdot 8^03⋅83+7⋅82+7⋅81+7⋅80 3⋅512+7⋅64+7⋅8+7⋅13 \cdot 512 + 7 \cdot 64
+ 7 \cdot 8 + 7 \cdot 13⋅512+7⋅64+7⋅8+7⋅1 1536+448+56+7=20471536 + 448 + 56 +
7 = 20471536+448+56+7=2047
2. Answer: 204720472047

B. 120412041204:

1. Expand the octal number: 1⋅83+2⋅82+0⋅81+4⋅801 \cdot 8^3 + 2 \cdot 8^2 + 0 \cdot
8^1 + 4 \cdot 8^01⋅83+2⋅82+0⋅81+4⋅80 1⋅512+2⋅64+0⋅8+4⋅11 \cdot 512 + 2 \cdot 64
+ 0 \cdot 8 + 4 \cdot 11⋅512+2⋅64+0⋅8+4⋅1 512+128+0+4=644512 + 128 + 0 + 4 =
644512+128+0+4=644
2. Answer: 644644644

Question #4: Convert Decimal to Octal

A. 655366553665536:
 1. Divide by 8 repeatedly: 65536÷8=8192r065536 \div 8 = 8192 \quad
r065536÷8=8192r0 8192÷8=1024r08192 \div 8 = 1024 \quad r08192÷8=1024r0
1024÷8=128r01024 \div 8 = 128 \quad r01024÷8=128r0 128÷8=16r0128 \div 8 = 16 \
quad r0128÷8=16r0 16÷8=2r016 \div 8 = 2 \quad r016÷8=2r0 2÷8=0r22 \div 8 = 0 \
quad r22÷8=0r2
2. Write remainders from bottom to top: 200000200000200000
3. Answer: 200000200000200000

B. 765247652476524:

1. Divide by 8 repeatedly: 76524÷8=9565r476524 \div 8 = 9565 \quad

r476524÷8=9565r4 9565÷8=1195r59565 \div 8 = 1195 \quad r59565÷8=1195r5
1195÷8=149r31195 \div 8 = 149 \quad r31195÷8=149r3 149÷8=18r5149 \div 8 = 18 \
quad r5149÷8=18r5 18÷8=2r218 \div 8 = 2 \quad r218÷8=2r2 2÷8=0r22 \div 8 = 0 \
quad r22÷8=0r2
2. Write remainders from bottom to top: 225354225354225354
3. Answer: 225354225354225354

Question #5: Convert Hexadecimal to Decimal

A. 37FD37FD37FD:

1. Expand the hexadecimal number: 3⋅163+7⋅162+F⋅161+D⋅1603 \cdot 16^3 + 7 \cdot

16^2 + F \cdot 16^1 + D \cdot 16^03⋅163+7⋅162+F⋅161+D⋅160 Where F=15F =
15F=15 and D=13D = 13D=13. 3⋅4096+7⋅256+15⋅16+13⋅13 \cdot 4096 + 7 \cdot 256
+ 15 \cdot 16 + 13 \cdot 13⋅4096+7⋅256+15⋅16+13⋅1
12288+1792+240+13=1423312288 + 1792 + 240 + 13 =
1423312288+1792+240+13=14233
2. Answer: 142331423314233

B. 1B91B91B9:

1. Expand the hexadecimal number: 1⋅162+B⋅161+9⋅1601 \cdot 16^2 + B \cdot 16^1 + 9

\cdot 16^01⋅162+B⋅161+9⋅160 Where B=11B = 11B=11. 1⋅256+11⋅16+9⋅11 \cdot
256 + 11 \cdot 16 + 9 \cdot 11⋅256+11⋅16+9⋅1 256+176+256 + 176 +256+176+

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Question #6: Convert Decimal to Hexadecimal

A. 409540954095:

1. Divide by 16 repeatedly: 4095÷16=255r15 (F)4095 \div 16 = 255 \quad r15 \,

(F)4095÷16=255r15(F) 255÷16=15r15 (F)255 \div 16 = 15 \quad r15 \,
(F)255÷16=15r15(F) 15÷16=0r15 (F)15 \div 16 = 0 \quad r15 \, (F)15÷16=0r15(F)
2. Write remainders from bottom to top: FFFFFFFFF
3. Answer: FFFFFFFFF

B. 256192561925619:

1. Divide by 16 repeatedly: 25619÷16=1601r325619 \div 16 = 1601 \quad

r325619÷16=1601r3 1601÷16=100r11601 \div 16 = 100 \quad r11601÷16=100r1
100÷16=6r4100 \div 16 = 6 \quad r4100÷16=6r4 6÷16=0r66 \div 16 = 0 \quad
r66÷16=0r6
2. Write remainders from bottom to top: 641364136413
3. Answer: 641364136413

Question #7: Convert Binary to Hexadecimal

A. 110100011110111010001111011101000111101:

1. Group bits in sets of 4 from right to left: 1 1010 0011 11011 \, 1010 \, 0011 \,
11011101000111101 Add leading zeros if necessary: 0001 1010 0011 11010001 \,
1010 \, 0011 \, 11010001101000111101
2. Convert each group to hex: 0001=1, 1010=A, 0011=3, 1101=D0001 = 1, \, 1010 =
A, \, 0011 = 3, \, 1101 = D0001=1,1010=A,0011=3,1101=D
3. Answer: 1A3D1A3D1A3D

B. 111011101110111011101110:

1. Group bits in sets of 4 from right to left: 1110 11101110 \, 111011101110

2. Convert each group to hex: 1110=E, 1110=E1110 = E, \, 1110 = E1110=E,1110=E
3. Answer: EEEEEE

Question #8: Add Binary Numbers

A. 1011.1101+11.11011.1101 + 11.11011.1101+11.1:

1. Align numbers: 1011.1101+0011.10001011.1101 + 0011.10001011.1101+0011.1000

2. Perform binary addition: 1111.01011111.01011111.0101
3. Answer: 1111.01011111.01011111.0101

B. 10011011+1001110110011011 + 1001110110011011+10011101:

1. Align numbers: 10011011+1001110110011011 + 1001110110011011+10011101

2. Perform binary addition: 100110000100110000100110000
 3. Answer: 100110000100110000100110000

Question #9: Subtract Binary Numbers

A. 101011100−11101110101011100 - 11101110101011100−11101110:

1. Perform binary subtraction: 101011100−011101110101011100 -

011101110101011100−011101110
2. Borrow as necessary: 110110110
3. Answer: 110110110

B. 101010100−1111000101010100 - 1111000101010100−1111000:

1. Perform binary subtraction: 101010100−0001111000101010100 -

0001111000101010100−0001111000
2. Borrow as necessary: 100100100100100100100100100
3. Answer: 100100100100100100100100100

Question #10: Perform Operations

A. ABEBE+CADEFABEBE + CADEFABEBE+CADEF:

1. Convert to decimal:
o ABEBE=703038ABEBE = 703038ABEBE=703038
o CADEF=831471CADEF = 831471CADEF=831471
2. Add decimal values: 703038+831471=1534509703038 + 831471 =
1534509703038+831471=1534509
3. Convert back to hex: 1534509=17788D1534509 = 17788D1534509=17788D
4. Answer: 17788D17788D17788D

B. (567)8−(473)8(567)_8 - (473)_8(567)8−(473)8:

1. Convert octal to decimal:

o (567)8=375(567)_8 = 375(567)8=375
o (473)8=315(473)_8 = 315(473)8=315
2. Subtract decimal values: 375−315=60375 - 315 = 60375−315=60
3. Convert back to octal: 60=(74)860 = (74)_860=(74)8
4. Answer: (74)8(74)_8(74)8