Math 55,
First Midterm Exam
SOLUTIONS
(1) There are twelve positive integers less than 36 that are relatively prime to 36. Hence
φ
(36) =
12
. Since 37 is a prime number, we have
φ
(37) = 37
−
1 =
36
. In the second
question of Quiz # 2 we saw that
φ
(
p
k
) =
p
k
−
p
k

1
for any power of a prime number
p
.
Therefore,
φ
(81) =
φ
(3
4
) = 3
4
−
3
3
=
54
and
φ
(1024) =
φ
(2
10
) = 2
10
−
2
9
=
512
.
(2) It is important to note that the domain is the set of nonnegative integers.
(a) The statement
∃
x
((
x
2
<
10)
∧
(

3
−
x

>
2)) is
true
because
x
= 0 satisFes both
inequalities in the conjunction.
(b) The statement
∀
x
((
x
n
= 4)
→
(
x
−
5
>
1)) is
false
. It does not hold for
x
= 3.
(c) The statement
∀
x
∃
y
(
x
+
y
= 0) is
false
because positive integers have no additive
inverses among the nonnegative integers.
(d) The statement
∃
x
∀
y
(
xy
= 0) is
true
because
x
= 0 is a nonnegative integer, and it
satisFes 0
·
y
= 0 for all
y
.
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 Spring '08
 STRAIN
 Math, Integers, Natural number, Prime number, positive integers

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