Prepared by Maya Ahmed
April 3, 2003
1.
Brualdi 2.4
Show that if
n
+ 1 integers are chosen from the set
{
1
,
2
,
3
, ..,
2
n
}
, then there
are always two which diﬀer by 1.
Answer:
Partition the set
{
1
,
2
,
3
, ..,
2
n
}
into
n
subsets as following
{
1
,
2
}{
3
,
4
}
, ....
,
{
2
n

1
,
2
n
}
. Now if we choose
n
+ 1 distinct numbers from
{
1
,
2
,
3
, ..,
2
n
}
, then two must come from the same subset by pigeonhole princi
ple. These two numbers must then diﬀer only by 1.
2.
Brualdi 2.5
Show that if
n
+ 1 integers are chosen from the set
{
1
,
2
,
3
, ..,
3
n
}
, then there
are always two which diﬀer by at most two.
Answer:
Partition the set
{
1
,
2
,
3
, ..,
3
n
}
into
n
subsets as following
{
1
,
2
,
3
}
,
{
4
,
5
,
6
}
, ....
,
{
3
n

2
,
3
n

1
,
3
n
}
. Now if we choose
n
+ 1 distinct
numbers from
{
1
,
2
,
3
, ..,
3
n
}
, then two must come from the same subset by
pigeonhole principle. These two numbers must then diﬀer by no more than
two.
3.
Brualdi 2.8
Use the pigeonhole principle to prove that the decimal expansion expansion of
a rational number
m/n
eventually is repeating.
Answer:
Assume
m < n
, since if
m > n
then after dividing
m
by
n
we get
m
=
pn
+
r
and the proof will apply to
r/n
.
Let
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 Spring '10
 CHASTKOFSKY
 Combinatorics, Decimal, Natural number, Prime number

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