Math 366 Practice Problems for Exam 2
Autumn 2007
1.Constructive Proof of an Existential Statement.
(a) Prove that there is an even integer
n
such that
n mod
3 = 1.
(b) Prove that there exists a rational number
q
such that 9
q
2
= 4.
(c) Prove that there exist two real numbers whose product is less than their sum.
(d) Prove that there exist two real numbers which are not equal to each other and whose product
is equal to their sum.
(e) Prove that there is an odd integer
n
such that
n >
1 and
n
has the form 3
k
+ 1 for some
integer
k
.
2. Direct Proof of a Universal Statement.
(a) Prove that if
n
is an integer which is divisible by 6 then
n
is divisible by 3.
(b) Prove that for any integers
a, b, c,
and
d
, if
a
divides
b
and
c
divides
d
then
a
·
c
divides
b
·
d
.
(c) Prove that the product of two odd integers is odd.
(d) Prove that if
n
is an integer which is divisible by 5 then 3
n
is divisible by 15.
(e) Prove that for any nonzero rational numbers
a
and
b
there is a rational number
x
such that
ax
+
b
= 0.
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 Winter '08
 JOSHUA
 Math, Mathematical Induction, Rational number

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