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Section 7.1
183
(b) Writing
n
as the product of prime powers as in part (a), the product
(al
+
1)(a2
+
1)···
(as
+
1)
is odd if and only if each factor
ai
+
1 is odd, hence, if and only if each
ai
=
2b
i
is even. And this
occurs if and only if
n
=
(p~lp~2
...
p~8)2
is a perfect square.
(c) At the end of the day, a picket is white if and only if it has a number which is a perfect square.
To see why, we need to determine those pickets whose color changes an even number of times.
Picket
k
changes color each time there is an integer
d,
1 <
d
S;
k,
which divides
k.
The number
of such
d,
together with 1, is the total number of factors of
d.
By (a), this number is odd (and the
number of
d
>
1 is even) if and only if
k
is a perfect square.
Exercises 7.1
1.
[BB] 13 . 12 . 11 .
. ·6
=
P(13,
8)
=
51,891,840
2. 8· 7· 6
=
P(8,
3)
=
336
3. [BB]
10·9·8·7
=
P(1O,4)
=
5040
4. In 30 . 29
... 23
=
P(30,
8) ways.
5. [BB] 3 x 5!
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 Summer '10
 any
 Graph Theory

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