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Section
4.3
71"(500)
=
95,
1;~gO ~
80.456 and
50~)~~OJoo ~
1.181.
13. There are approximately
5000/ In 5000
~
587 primes less than 5000,
50,000/ In 50,000
~
4621 primes less than 50,000,
500,000/In500,000
~
38,103 primes less than 500,000 and
5,000,000/ In 5,000,000
~
324,150 primes less than 5,000,000.
85
14. [BB] This is a special case of Exercise 11, Section 4.2 since if p and
q
are distinct primes, then p and
q
are relatively prime. The result also follows quickly from the Fundamental Theorem of Arithmetic
since, writing
n
=
P1P2
...
Pr,
the hypotheses say that one of the Pi is P and some other
Pi
is
q.
15. (a) Since
2xo
+
5yo
=
2x
+
5y,
we have
5(yo

y)
=
2(x

xo).
Since 5 divides the left side,
5
I
2(x

xo),
so 5
I
(x

xo)
by Proposition 4.3.7. Thus
x

xo
=
5k
(hence
x
=
xo
+
5k)
for
some integer
k.
From
5(yo

y)
=
2(x

xo),
we conclude that
5(yo

y)
=
10k,
so
Yo

Y
=
2k
and
y
=
Yo

2k
as desired.
(b) This follows from part (a) since
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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
 Graph Theory

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