MA 315, Spring 2011
Homework 2
Due Wednesday, February 2, 2011
Your paper must be typed. Each proof must be written in complete sentences with
correct spelling, punctuation and grammar.
(1) Prove the following: If
n
is an odd number then
n
2

1 is divisible by 8.
Proof:
Let
n
be an odd integer.
Then there exists an integer
k
such that
n
= 2
k
+ 1. Therefore
n
2

1 = (2
k
+ 1)
2

1 = 4
k
2
+ 4
k
= 4
k
(
k
+ 1). This
previous line says that
n
2

1 is divisible by 4, we now show that
n
2

1 is
actually divisible by 8. Note that
k
can be either even or odd; we will show
that, in either case,
n
2

1 is divisible by 8. If
k
is even, then
k
= 2
m
for some
integer
m
, therefore
n
2

1 = 4
·
2
m
(
k
+ 1) = 8
m
(
k
+ 1), which shows that
8

n
2

1, so
n
2

1 is divisible by 8. If
k
is odd, then
k
+1 is even, so
k
+1 = 2
m
for some integer
m
, so
n
2

1 = 4
k
·
2
m
= 8
km
which shows that 8

n
2

1, so
n
2

1 is divisible by 8. This yields that, whether
k
is even or odd,
n
2

1 is
divisible by 8.
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 Spring '11
 PeterTurbek
 Addition, Prime number, Elementary number theory, Parity, Evenness of zero

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