MATH 135
Fall 2007
Assignment #5
Due: Wednesday 24 October 2007, 8:20 a.m.
HandIn Problems
1. (a) Convert (84 808)
9
to base 10. Show your work.
(b) Convert 37 390 to base 16, using
A
for ten,
B
for eleven,
C
for twelve,
D
for thirteen,
E
for fourteen, and
F
for ﬁfteen. Show your work.
2. Add and multiply the numbers (3157)
8
and (445)
8
, expressing your answers in base 8.
3. (a) Determine the prime factorizations of 39 204 000 and 9 922 500.
(b) Determine gcd(39 204 000
,
9 922 500) and lcm(39 204 000
,
9 922 500).
(c) Determine the number of positive divisors of 9 922 500.
(d) Determine the number of positive common divisors of 39 204 000 and 9 922 500 that are
multiples of 10.
4. A positive integer
n
is called a perfect square if
n
=
m
2
for some positive integer
m
.
(a) If
n
is a perfect square and
n
has prime factorization
n
=
p
c
1
1
· · ·
p
c
k
k
, prove that
c
1
,c
2
,...,c
k
are all even.
(b) Suppose that
a,b,c
∈
P
with
a,b,c >
1 and
ab
=
c
2
. If gcd(
a,b
) = 1, prove that
a
and
b
are both perfect squares.
5. Suppose
a,b
∈
Z
.
(a) Prove that if
a

b
, then
a
3

b
3
.
(b) Suppose
a
=
p
c
1
1
· · ·
p
c
k
k
and
b
=
p
d
1
1
· · ·
p
d
k
k
for some primes
p
1
,...,p
k
and some nonnegative
integers
c
1
,...,c
k
,d
1
,...,d
k
. Prove that if
a
3

b
3
, then
a

b
.
6. (a) Determine the number of zeroes with which the base 10 representation of 135! ends.
Explain how you got your answer.
(b) Determine the number of zeroes with which the base 16 representation of 135! ends.
Explain how you got your answer.
7. Suppose that
x
is a digit in base 16 and
y
is a digit in base 7. If (
xx
)
16
= (2
y
0)
7
, determine
x
and
y
.
8. Every polyhedron obeys Euler’s Formula:
V

E
+
F
= 2, where
V
is the number of vertices,
E
is the number of edges and
F
is the number of faces.
A polyhedron has 20 faces that are hexagons,
p
faces that are pentagons, and no additional
faces. Three faces meet at each vertex (that is, three edges meet at each vertex).
(a) In terms of
p
, determine the number of edges that the polyhedron has.
(b) In terms of
p
, determine the number of vertices that the polyhedron has.
(c) Determine the value of
p
.
(This problem is not directly related to the course material, but is included to keep your problem
solving skills sharp.)