MATH 127: Exam 3 Review
Monday, April 18, 2011
1. Show that the square of every odd integer is of the form 8
k
+ 1
2. Show that if
a, b
∈
Z
, not both 0, and
c
∈
Z
with
c
6
= 0 then (
ca, cb
) =

c

(
a, b
)
3. Show that if
k
∈
N
then (3
k
+ 2
,
5
k
+ 3) = 1
4. Use the Euclidean Algorithm to find the gcd of 1776 and 1492 and then express this gcd as
a linear combination of 1776 and 1492
5. Show that if
a, b
∈
N
and
a
3

b
2
then
a

b
6. For
p
prime, we say
p
a
strongly divides
n
, denoted
p
a

n
iff
p
a

n
but
p
a
+1
6 
n
(a) Show that if
p
a

m
and
p
b

n
then
p
a
+
b

n
(b) Show that if
p
a

m
and
p
b

n
with
a
6
=
b
then
p
min
(
a,b
)

(
m
+
n
)
7. Show that if
a, b, c
∈
Z
and
c

ab
then
c

(
a, c
)(
b, c
)
8. Show that if
a, b, c
∈
N
with (
a, b
) = 1 and
ab
=
c
n
then there exists
d, e
∈
N
with
a
=
d
n
,
b
=
e
n
9. For the following diophantine equations, either find all solutions or show that there are none
(a) 30
x
+ 47
y
=

11
(b) 25
x
+ 95
y
= 970
10. A vegetarian banquet is offering a seitan dish and a tofu dish. The seitan costs $11 and the
tofu costs $8. What can you conclude if the total bill is:
(a) $96
(b) $69
11. Let
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 Spring '07
 GHEORGHICIUC
 Math, Remainder, Equivalence relation, Euclidean algorithm, Congruence relation

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