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homework23sol - 2004 Summer COT 5907 Homework #2,3...

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2004 Summer COT 5907 Homework #2,3 Solutions Question 1: The first five prime numbers greater than 30 10 are: 1000000000000000000000000000057 1000000000000000000000000000099 1000000000000000000000000000211 1000000000000000000000000000231 1000000000000000000000000000271 Code attached. Question 2: ) 101 29 17 13 ( ) 187072301 ( 3 × × × = φ ) 101 1 1 ( ) 29 1 1 ( ) 17 1 1 ( ) 13 1 1 ( 187072301 - × - × - × - × = 101 100 29 28 17 16 13 12 187072301 × × × × = 100 28 16 12 17 2 × × × × = 155366400 = Code attached. Question 3: 12x 37 mod 91 5x 62 mod 72 11x 3 mod 25 => x 37*38 mod 91 41 mod 91 x 62*29 mod 72 70 mod 72 x 3*16 mod 25 23 mod 25 Use the Chinese Remainder Threom: x = 102598 (mod 163800), thus all possible solutions are x {102598 + 163800n | n Z} Code attached.
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Question 4: Primitive roots and corresponding x: Primitive root: 2; x=108 Primitive root: 3; x=66 Primitive root: 7; x=102 Primitive root: 17; x=162 Primitive root: 22; x=96 Primitive root: 29; x=132 Primitive root: 35; x=198 Primitive root: 39; x=114 Primitive root: 41; x=204 Primitive root: 48; x=114 Primitive root: 57; x=24 Primitive root: 72; x=72 Primitive root: 75; x=144 Primitive root: 85; x=138 Primitive root: 91; x=36 Primitive root: 92; x=96 Primitive root: 106; x=102 Primitive root: 108; x=198 Primitive root: 112; x=36 Primitive root: 116; x=18 Primitive root: 118; x=6 Primitive root: 127; x=12 Primitive root: 130; x=174 Primitive root: 131; x=78 Primitive root: 133; x=186 Primitive root: 141; x=144 Primitive root: 142; x=192 Primitive root: 145; x=78 Primitive root: 149; x=48 Primitive root: 152; x=204 Primitive root: 155; x=54 Primitive root: 158; x=156 Primitive root: 159; x=18 Primitive root: 160; x=174 Primitive root: 162; x=156 Primitive root: 164; x=72 Primitive root: 165; x=24 Primitive root: 166; x=66
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Primitive root: 167; x=162 Primitive root: 174; x=186 Primitive root: 175; x=6 Primitive root: 181; x=108 Primitive root: 187; x=48 Primitive root: 191; x=192 Primitive root: 195; x=132 Primitive root: 202; x=138 Primitive root: 205; x=12 Primitive root: 207; x=54 Number of primitive roots found:
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homework23sol - 2004 Summer COT 5907 Homework #2,3...

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