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UIUC Putnam Training Sessions
Fall 2010
Putnam Training Session 6: Number Theory
•
Primes and Composite Numbers:
An integer
n
≥
2 is called
prime
if its only positive divisors are 1 and
n
; and is called
composite
otherwise. Equivalently, an integer
n
≥
2 is prime if it cannot be
written in the form
n
=
ab
with integers
a,b
≥
2. (Note that only
integers
≥
2 are classiﬁed as prime or composite; in particular, the
number 1 is neither prime nor composite.)
•
Congruences:
Let
a,b
∈
Z
, and
m
∈
N
. We say
“
a
is congruent to
b
modulo
m
”
, and write “
a
≡
b
mod
m
”, if
m
divides
a

b
. The integer
m
is called the
modulus
of the congruence.
•
Properties of congruences
What makes congruences so useful is
that, to a large extent, they can be manipulated like ordinary equa
tions. Congruences
to the same modulus
can be added, multiplied,
and taken to a ﬁxed positive integral power; i.e., for any
a,b,c,d
∈
Z
and
m
∈
N
we have:
1. If
a
≡
b
mod
m
and
c
≡
d
mod
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This note was uploaded on 01/31/2011 for the course COMP 101 taught by Professor Rajivsir during the Winter '10 term at National.
 Winter '10
 RajivSir

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