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Math 25: Advanced Calculus
Fall 2010
Homework # 1 – Solutions
Due Date:
Friday, October 1
A.5.2:
Show that there are inﬁnitely many prime numbers.
Proof:
We will use proof by contradiction. Suppose to the contrary that there are only
ﬁnitely many primes, and call them
p
1
,p
1
,...,p
r
. Consider the number
N
=
p
1
·
p
2
·
...
·
p
r
+ 1
.
Since
N
÷
p
i
has remainder 1 for all of our primes
p
i
, we see that
N
is not divisible by
any prime
p
i
. We know, however, that any natural number can be written as a product
of prime numbers. This contradicts our assumption that
p
1
,...,p
r
are the only prime
numbers.
±
A.6.1:
Prove the following assertion by contraposition: If
x
is irrational, then
x
+
r
is
irrational for all rational numbers
r
.
Proof:
We want to show that
P
⇒
Q
, where
P
:
x
is irrational
Q
:
x
+
r
is irrational for all rational numbers
r
.
In order to employ proof by contraposition, we need to show that
∼
Q
⇒∼
P
, and
we begin by negating the statements
P
and
Q
:
∼
Q
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 Fall '10
 StevenKlee
 Calculus, Prime Numbers

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