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Section 4.1
2. (a) [BB] True. If
a
and
b
are real numbers, certainly
a

b
is a real number.
(b) False. If
a
=
1,
b
=
2, then
a

b
=
1 but
b

a
=
1.
75
(c) False. Ifa
=
1,b
=
2,c
=
3, then
(ab)
c
=
13
=
4, while
a(bc)
=
1(1)
=
2.
3. Let
a
=
3,
b
=
4, c
=
2. Then 3 :5 4 but 3(
2)
=
6
1:.
8
=
4( 2).
4.
(a) [BB]
q
=
29, r
=
7;
(b) [BB]
q
=
30, r
=
10;
(c) [BB]
q
=
29, r
=
7;
(d) [BB]
q
=
30,
r =
10.
5.
(a)
q
=
278,
r =
4;
(d)
q
=
279, r
=
15;
6. (a)
q
=
316, r
=
21;
(d)
q
=
20, r
=
2984;
(b)
q
=
279,
r =
15;
(e)
q
=
0, r
=
19;
(b)
q
=
29, r
=
972;
(e)
q
=
55, r
=
134;
(c)
q
= 278,
r
=
4;
(0
q
=
1,
r =
5267.
(c)
q
=
70, r
=
613;
(0
555,555,123
=
555,555,555  432
=
5(111,111,111) 
432
=
4(111,111,111) + 111,111,111  432
=
4(111,111,111) + 111,110,679
So
q
=
4; r
=
111,110,679
(g)
q
=
2,104,000; r
=
0
7. [BB] Write
a
=
3q
+ r with r
=
0,1,2.
Case 1:
r
=
O. In this case,
a
=
3q,
so
a
2
=
9q2
=
3k
with
k
=
3q2.
Case 2:
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Real Numbers, Graph Theory

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