Discrete Structures
Oct 24 2011
Assignment 8:
Due at beginning of class Monday, Oct 24th
Prof. Hopcroft
*******Note: this homework is subject to point evaluations for clarity of writing and clarity
of mathematical statements.********
*******Please print out and staple the grade sheet to the back of your homework!******
1. Consider the integers,
I
=
{
...,

2
,

1
,
0
,
1
,
2
,...
}
, and a prime
p.
(a) If,
i
∈
I
, and
i
≥
0 what is
i
mod
p
?
Solution
We note that any number
i
≥
0 can be written as
i
=
ap
+
r
, for some positive
a
and
r
∈
[0
,p

1]. In particular
a
will be
j
i
p
k
. Once we have
i
in this form, it is easy to see
i
mod
p
≡
r
mod
p
.
(b) If
i
∈
I
, and
i <
0 what is
i
mod
p
?
Solution
As above, we write
i
=
ap
+
r
. If we can write
i
in such a form for
r
∈
[0
,p

1], then the
answer will be the same. The catch, is now that
i <
0,
a
will be negative. Interestingly enough,
a
will still be
j
i
p
k
. Try it out!
2. (a) Prove that modular arithmetic under
p
, ie mod
p
, is an equivalence relation.
Solution
For a relation to be an equivalence relation, it must be reﬂexive, transitive, and sym
metric.
Symmetric