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Congruences in Number Theory
Jonathan L.F. King
University of Florida, Gainesville FL 326112082, USA
squash@math.ufl.edu
Webpage
http://www.math.uﬂ.edu/
∼
squash/
30 April, 2006
(at
22:02
)
Entrance.
Let Primes(
N
) mean the set
of primes
that divide
N
.
An
arithmetic progression
means a set
T
+
G
Z
of integers, where the
gap
G
is
a posint and
translation
T
an integer. Use
comb
,
also, for “arithmetic progression”.
A comb
C
:=
T
+
G
Z
is
coprime
if
T
⊥
G
.
Divisibility Conundra
Here is a soln to LeVeque’s
#
7
P
63
.
Short soln.
Fix a coprime comb
T
+
G
Z
and
posint
N
.
Let
F
be the maximum factor of
N
st.
F
⊥
G
. Letting
Q
:=
N
F
, then,
Primes(
Q
)
⊂
Primes(
G
)
.
1
:
Since
F
⊥
G
, the
CRT
applies to produce an integer
x
with
x
≡
G
T
and
x
≡
F
1
.
2
:
So in order to show that
x
⊥
N
, we need show that
x
⊥
Q
. FTSOC, suppose
p
is a prime with
p
•
x
and
p
•
Q
. This latter forces
p
•
G
, by (1). Now LhS(2)
forces
T
•
p
. This contradicts that
T
⊥
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This note was uploaded on 01/26/2012 for the course MAC 3472 taught by Professor Jury during the Fall '07 term at University of Florida.
 Fall '07
 JURY
 Calculus, Number Theory, Congruence

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