Introduction
Primes
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics  Primes
302
Integers
“
God made the integers;
all else is the work of man
”
Leopold Kroenecker
Discrete Mathematics  Primes
303
Most of useful properties of integers are related to division
If
a
and
b
are integers with
a
≠
0,
we say that
a
divides
b
if
there is an integer
c
such that
b = ac.
When
a
divides
b
we say that
a
is a
divisor
(
factor
)
of
b,
and
that
b
is a
multiple
of
a.
The notation
a  b
denotes that
a
divides
b.
We write
a  b
when
a
does not divide
b
Example.
Let
n
and
d
be positive integers. How many positive
integers not exceeding
n
are divisible by
d?
The numbers in question have the form
dk,
where
k
is a positive
integer and
0 < dk
≤
n.
Therefore,
0 < k
≤
n/d.
Thus the answer is
n/d
Division
∕
Discrete Mathematics  Primes
304
Properties of Divisibility
Let
a,
b,
and
c
be integers. Then
(i)
if
a  b
and
a  c,
then
a  (b + c);
(ii)
if
a  b,
then
a  bc
for all integers
c;
(iii)
if
a  b
and
b  c,
then
a  c.
Proof.
(i)
Suppose
a  b
and
a  c.
This means that there are
k
and
m
such that
b = ak
and
c = am.
Then
b + c = ak + am = a(k + m),
and
a
divides
b + c.
Discrete Mathematics  Primes
305
Properties of Divisibility
(cntd)
If
a,
b,
and
c
are integers such that
a  b
and
a  c,
then
a  mb + nc
whenever
m
and
n
are integers.
Proof.
By part
(ii)
it follows that
a  mb
and
a  nc.
By part
(i) it follows that
a  mb + nc.
If
a  b
and
b  a,
then
a =
±
b.
Proof.
Suppose that a  b
and
b  a.
Then
b = ak
and
a = bm
for some
integers
k
and
m.
Therefore
a = bm = akm,
which is possible only if
k,m =
±
1.
Discrete Mathematics  Primes
306
The Division Algorithm
Theorem
(The division algorithm)
Let
a
be an integer and
d
a positive integer.
Then there are
unique integers
q
and
r,
with
0
≤
r < d,
such that
a = dq + r
d
is called the
divisor
,
a
is called the
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 Spring '10
 AndreiBulatov
 Prime number, Goldbach

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