Math 3221 Number Theory - Homework Until Test 2 Worksheet With Answers - Philipp Braun Page 3
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16Section 3.2
page 49, 1. Determine whether the integer 701 is prime by testing all primes p
701 as possible
divisors. Do the same for integer 1009.
Solution.
701
26.47, i.e. we have to test the primes 2, 3, 5, 7, 11, 13, 17, 19 and 23 as
possible divisors of 701:
701/2 = 350.5 / Z
2 701.
701/3 = 233.6 / Z
3 701.
701/5 = 140.2 / Z
5 701.
701/7
100.14 / Z
7 701.
701/11 = 63.72 / Z
11 701.
701/13
53.92 / Z
13 701.
701/17
41.24 / Z
17 701.
701/19
36.89 / Z
19 701.
701/23
30.48 / Z
23 701. Hence 701 is a prime.
Since
1009
31.76, we have to test the primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 and 31 as
possible divisors of 1009:
1009/2 = 504.5 / Z
2 1009.
1009/3 = 336.3 / Z
3 1009.
1009/5 = 201.8 / Z
5 1009.
1009/7
144.14 / Z
7 1009.
1009/11 = 91.72 / Z
11 1009.
1009/13
77.62 / Z
13 1009.
1009/17
59.35 / Z
17 1009.
1009/19
53.11 / Z
19 1009.
1009/23
43.87 / Z
23 1009.
1009/29
34.79 / Z
29 1009.
1009/31
32.54 / Z
31 1009. Hence 1009 is a prime.
2. Employing the Sieve of Eratosthenes, obtain all the primes between 100 and 200.
Solution.
¨ ¨
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100 101
102 103
104
105
106 107
108 109
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¨ ¨
¨ ¨
¨ ¨
¨ ¨
¨ ¨
r r
110
111
112 113
114
115
116
117
118
119
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r r
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¨ ¨
¨ ¨
¨ ¨
¨ ¨
¨ ¨
120
121
122
123
124
125
126 127
128
129
¨ ¨
¨ ¨
r r
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¨ ¨
¨ ¨
¨ ¨
130 131
132
133
134
135
136 137
138 139
¨ ¨
¨ ¨
¨ ¨
r r
¨ ¨
¨ ¨
¨ ¨
¨ ¨
¨ ¨
140
141
142
143
144
145
146
147
148 149
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¨ ¨
¨ ¨
¨ ¨
¨ ¨
¨ ¨
¨ ¨
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150 151
152
153
154
155
156 157
158
159
¨ ¨
r r
¨ ¨
¨ ¨
¨ ¨
¨ ¨
¨ ¨
r r
160
161
162 163
164
165
166 167
168
169
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¨ ¨
¨ ¨
¨ ¨
¨ ¨
¨ ¨
170
171
172 173
174
175
176
177
178 179
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