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
m
, then
a
+
c
≡
b
+
d
mod
m
.
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 Winter '10
 RajivSir
 Prime number, decimal digits, mod m., UIUC Putnam Training, Putnam Training Sessions

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