Section 36
1. Working in
Z
10
(a)
3 + 3 = 6
(b)
6 + 6 = 2
(c)
7 + 3 = 0
(d)
9 + 8 = 7
(e)
12 + 4 =?
Technically, according to the de°nition of
Z
n
on page
310, the number
12
is not in
Z
10
, so this addition is not de°ned.
Personally, in the context of
Z
10
, I think of
12
as being equal to
2
which is in
Z
10
. So
12 + 4 = 6
.
(f)
3
°
3 = 9
(g)
4
°
4 = 6
(h)
7
°
3 = 1
(i)
5
°
2 = 0
(j)
6
°
6 = 6
(k)
4
°
6 = 4
(l)
4
°
1 = 4
(m)
12
°
5 = 0
, or
12
°
5
is not de°ned (see part
(
e
)
)
(n)
5
±
8 = 7
. I±m sure that
±
3
is not an acceptable answer for this
textbook because he does not consider
±
3
to be in
Z
10
. Lots of
other people would not accept
±
3
, and I don±t either. What is the
point of this problem if
±
3
is an acceptable answer? One could
have just as well have answered
21
to part
(
h
)
.
(o)
8
±
5 = 3
(p)
8
=
7 = 4
because
4
°
7 = 28 = 8
(q)
5
=
9 = 5
because
5
°
9 = 45 = 5
2. Solve for
x
(a)
3
x
= 4
in
Z
11
. The answer is
x
= 5
.
(b)
4
x
±
8 = 9
in
Z
11
becomes
4
x
= 17 = 6
, and the answer is
x
= 7
.
1
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(c)
3
x
+ 8 = 1
in
Z
10
becomes
3
x
=
±
7 = 3
, and the answer is clearly
x
= 1
.
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 Spring '08
 STAFF
 Prime number, Hebrew numerals, Euclidean algorithm, Z10

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