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98
Solutions to Exercises
7. (a) [BB] The given rule is w· a = 0 (mod 11) for a =
(al,a2,
... , alO)
and
w = (1,1,1,1,1,1,1,1,1,1).
(b) Suppose an error is made in the ith digit of an ISBN number so that instead of
ala2
...
ai
...
alO,
the number appears
as
ala2'
..
a~
...
alO,
with
a~
:F
ai.
Then 0 =
at
+
a2
... +
ai
+
... +
alO
¢
at
+
a2
+
... +
a~
+
... +
alO
because
[at
+
a2
+
... +
ai
+
... + alOl 
[at
+
a2
+
... +
a~
+
... + alOl
=
ai

a~
¢ 0
(mod 11)
since
ai
and
a~
are
digits between 0 and 9.
(c) Suppose
ata2
... alO is a correct ISBN number. The transposition of
ai
and aHt cannot be
detected because when we apply our test to the incorrect number
at
... aH
t
ai
...
alO,
we compute
at
+
... +
aiH
+
ai
+
... + alO which is, of course, congruent to 0 (mod 11) because it's the
exactly the same as
at
+
... +
ai
+ aHt +
... +
alO.
8.
(a)[BB]
113579024688;
(d)
[BB]
063042006355;
(b)
102468135796;
(e)
063000164257;
(c) 062608393496;
(0 8294471 04931 .
9. (a) [BB] Not valid: 3(0 + 2 + 4 + 6 + 8 + 0) + (1 + 3 + 5 + 7 + 9 + 1) = 60 + 26 = 86 ¢ 0
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 Summer '10
 any
 Graph Theory

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