Section
4.4
91
Exercises 4.4
1.
(a)
[BB]
5,12,19,
2,
9,
16
are in
5.
4,11,18,
3,
10,
17
are in
3.
(b) [BB] The general element
of
5
is an integer
of
the form
7k
+
5 for some integer
k.
The general
form
of
an element in
3
is an integer
of
the form
7k

3 for some integer
k.
2.
(a)
3,16,29,42,
10,
23, 36,
49,
62
are elements
of
3.
11,24,37,50,
2,
15,
28, 41,
54
are elements
of
2.
(b) The general integer in
3
is
of
the form
13k
+
3 for some integer
k.
The general integer in
2
is
of
the form
13k

2 for some integer
k.
3.
(a) [BB]
1286 = 32(39)
+
38,
so
1286
(mod
39) =
38.
(b)
43,197 = 129(333)
+
240,
so
43,197
(mod
333) = 240.
(c) [BB]
545,608
= 10,699(51)
+
41,
so
545,608
(mod
51) =
41.
(d)
125,617
= 399(315)
+
68,
so
125,617
(mod
315) =
68.
(e)
11,111,111,111 = 10,001,000(1111)
+
111,
so
11,111,111,111
(mod
1111) = 111.
4.
(a)
[BB] False:
18

2 = 16
is not divisible by
10.
(c) [BB] True:
44

(8)
=
52
is divisible by
13.
(b) True:
13

7
=
20
is divisible by
5.
(d) False:
423 
17 = 406
is divisible by
29,
so
17 = 423.
(e) False:
400 
(18)
= 418
is divisible by
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 Summer '10
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 Graph Theory, Self number, Prime number

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