MATH 135
Fall 2007
Assignment #4
Due: Thursday 11 October 2007, 8:20 a.m.
HandIn Problems
1. Which of the following linear Diophantine equations have solutions? In each case, explain
brieﬂy why or why not. If there is a solution, determine the complete solution.
(a) 28
x
+ 91
y
= 40
(b) 2007
x

897
y
= 15
2. (a) Determine all nonnegative integer solutions to the linear Diophantine equation
133
x
+ 315
y
= 98000.
(b) Determine all nonnegative integer solutions to the linear Diophantine equation
133
x
+ 315
y
= 98000 with the additional property that
x
≤
640

2
y
.
3. Find the smallest positive integer
x
so that 141
x
leaves a remainder of 21 when divided by 31.
4. Suppose
a, b, c
∈
Z
. Prove that gcd(
a, c
) = gcd(
b, c
) = 1 if and only if gcd(
ab, c
) = 1.
5. Suppose
a, b, n
∈
Z
. Prove that if
n
≥
0, then gcd(
an, bn
) =
n
·
gcd(
a, b
).
6. Suppose that
f
(
x
) is a polynomial of degree
n
with coeﬃcients from
Q
and suppose that
g
(
x
) =
x

c
for some
c
∈
Q
.
(a) When
f
(
x
) is divided by
g
(
x
), we obtain a quotient
q
(
x
) and a remainder
r
(
x
). (See
Assignment #3.) Explain why, in this case,
r
(
x
) is a constant
r
(
x
) =
r
∈
Q
.
(b) Prove that
r
=
f
(
c
).
7. Consider the system of equations
a
+
b
= 2
m
2
b
+
c
= 6
m
a
+
c
= 2
Determine all real values of
m
for which
a
≤
b
≤
c
.
(This problem is not directly related to the course material, but is included to keep your problem
solving skills sharp.)
Recommended Problems
1. Text, page 50, #42
2. Text, page 51, #44
3. Text, page 51, #48
4. Text, page 52, #75
5. Text, page 52, #79
...continued