Section 4.3
83
am
+
bn
=
1 implies
am
=
bn
+
1,
so
(a
+
n)m
=
(b
+
m)n
+
1. Set
a'
=
a
+
n
and
b'
=
b
+
m.
Because 0
<
b
<
m,
0
<
b'
<
m
also. Because
n
<
a
<
0, 0
<
a'
<
n.
The uniqueness
of
a'
and
b'
follows from the uniqueness
of
a
and
b.
(Uniqueness can also be
proved directly as in (i).)
Exercises
4.3
1.
(a) [BB] 157 is prime;.
(b) [BB] 9831 is not prime; 3
I
9831;
(c) 9833 is prime;
(d) 55,551,111 is not prime; 11155,551,111.
(e) 2
216
,090 
1 is not prime: 2
216
,090 
1
=
(2
108
,045
+
1)(2
108
,045 
1).
2.
No. For example, take
n
=
9. Then
n
has no prime.factor smaller than
v'9.
3.
(a) [BB] Note that
n
=
p(nlp).
Now
if
nip
is not prime, then it has a prime factor
q
<
Jnlp
by
Lemma
4.3.4.
Since a prime factor
of
nip
is a prime factor
of
n,
q
would then be a prime factor
of
n
not exceeding
Jnlp,
contradicting the fact that the smallest prime factor
of
n
is bigger than
this.
(b) [BB] Note that 16,773,121
=
433(38,737) and that .../38,737
~
197. Since 433, the smallest
prime dividing 38,737, is larger than 197, 38,737 is prime, by
part
(a). Thus, 16,773,121
=
433(38,737) is the representation
of
16,773,121 as the product
of
primes.
4.
(a)
[BB] 856
=
2 . 2 . 2 . 107
=
2
3
.
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 Summer '10
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 Graph Theory, Prime number, Prime factor

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