1.
[20 points]
Let
a
,
b
,
c
∈
Z
with (
a, b
) = 1.
Prove (without
assuming the Fundamental Theorem of Arithmetic) that if
a

c
and
b

c
then
ab

c
.
HINT
: Since (
a, b
) = 1 there are integers
x
,
y
such that
ax
+
by
= 1.
2.
[20 points]
Let
a
,
b
,
c
∈
Z
and assume that
a
and
b
are not both
zero and let
d
= (
a, b
). Prove that
ax
+
by
=
c
for some integers
x
and
y
if and only if
d

c
.
HINT
: Use the result that
(
a, b
) = min
{
ma
+
nb
:
m, n
∈
Z
, ma
+
nb >
0
}
.
3.
[5 + 15 = 20 points]
(i) Define
pseudoprime
.
(ii) Prove that 161038 = (2)(73)(1103) is a pseudoprime.
4.
[5 + 15 = 20 points]
(i) State Euler’s Theorem.
(ii) Let
m
,
n
be positive relatively prime integers. Prove that
m
φ
(
n
)
+
n
φ
(
m
)
≡
1
(mod
mn
)
.
5.
[4
×
5 = 20 points]
(i) Define what it means for an arithmetic
function to be
multiplicative
.
(ii) Prove that if
f
is a multiplicative function and
f
(1) = 0 then
f
(
n
) = 0 for all
n
.
(iii) Prove that if
f
is a multiplicative function and
f
(1)
6
= 0 then
f
(1) = 1.
(iv) Suppose
f
(
n
) and
g
(
n
) are multiplicative functions.
Prove that
h
(
n
) =
f
(
n
)
g
(
n
) is multiplicative.
6.
[15 + 5 = 20 points]
(i) Let
m
,
n
be positive integers with
m

n
. Prove that
φ
(
m
)

φ
(
n
)
.
(ii) Prove or disprove the converse of (i).
1
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7.
[20 points]
Let
n
∈
Z
with
n >
1.
If
f
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 Spring '10
 BOYLAND
 Prime number, Multiplicative function, Law of Quadratic Reciprocity

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