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84
Solutions to Exercises
in the decomposition of
ac
as the product of primes are all even, that is, that
ac
=
p~O:l p~O:2
...
p~O:r
.
It
would follow that
ac
is the perfect square £2, where £
=
p~l
...
p~r.
Suppose then that
pO:
is the
largest power of a prime
p
dividing
ac
and that
p2/3
is the largest power of
p
dividing
b
2
,
then the largest
power of
p
dividing
(nm)2
is
po:+2f3.
Since this power is even,
a
is even.
It
follows that
ac
is a perfect
square, hence,
a
'" c.
8. (a) True. This is precisely Lemma 4.3.2.
(b) [BB] False. The prime described here has the miraculous property that it divides all natural num
bers greater than 1. Certainly this prime cannot be 2 since 2
~
3, neither can it be odd since if
p
is
an odd prime,
p
~
2.
9. (a) The range of
1
is the set of all prime numbers since any prime
p
is
I( n)
for some
n:
Take
n
=
p,
for example.
(b)
1
is not onetoone. For example,
1(2)
=
2
=
1(4)
but 2 # 4.
(c) Since the range is a proper subset of N,
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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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