Exercise
Set
4.1
*
In
13.
use the definitions
of
even, odd, prime, and
composite
to
justify
each
of
your
answers.
1.
Assume
that
k
is a particular integer.
a. Is
17
an odd integer?
b.
Is 0 an even integer?
c. Is
2k

1
odd?
2.
Assume
that m and
n
are particular integers.
a. Is 6m
+
8n
even?
b.
Is
lOmn
+
7
odd?
c. If m
>
n
>
0, is m
2

n
2
composite?
3.
Assume
that
rand
5
are
particular integers.
a.
Is
4r
5
even?
b.
Is
6r
+
45
2
+
3
odd?
c.
If
rand
5
are
both positive, is
r
2
+
2r
5
+
52
composite?
Prove the statements in
410.
4.
There
are integers m and
n
such that m
>
1 and
n
>
I and
I
1.
.
;; +
;;
IS
an mteger.
5.
There
are distinct integers m and
n
such that
1.
+
.!
is an
m
n
integer.
6.
There
are
real numbers
a
and
b
such that
.Ja
+
b
=
va
+.Jb.
7.
There
is an integer
n
>
5
such
that 2" 
1 is prime.
8.
There
is a real
number
x
such that
x
>
I and
2
x
>
x
10.
Definition:
An integer
n
is called a
perfect
square
if, and
only if,
n
=
k
2
for
some
integer
k.
9.
There
is a perfect square that can be written as a
sum
of
two
other
perfect squares.
10.
There
is an integer
n
such that
2n
2

5n
+
2 is prime.
Disprove the statements in
1113
by
giving a counterexample.
11.
For
all real numbers
a
and
b,
if
a
<
b
then
a
2
<
b
2
.
12.
For
all integers
n,
if
n
is odd then
n;
I is odd.
13.
For
all integers m and
n,
if
2m
+
n
is odd then m and
n
are
both odd.
In
1416,
determine
whether
the property is true for all integers,
true for no integers,
or
true for
some
integers and false for
other
integers. Justify
your
answers.
HIS.
_a"
=
(_a)"
16.
The
average
of
any two odd integers is odd.
Prove the statements in
17
and
18
by the method
of
exhaustion.
17. Every positive even integer less than
26
can be expressed
as a
sum
of
three
or
fewer perfect squares. (For instance,
10
=
1
2
+
3
2
and
16
=
4
2
.)
2
18.
For
each integer
n
with I
~
n
~
10,
n

n
+
II
is a prime
number.
19. a.
Rewrite the following theorem in three different ways: as
V
,
if
__
then
__
, as V
__
,
__
(with
out
using the words
if
or
then),
and as
If
__
, then
__
(without using an explicit universal quantifier).
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 Spring '08
 WOUTERS
 Algebra, Number Theory, Integers, Prime number, Ger

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