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Numbers and Polynomials
Homework 4
Let
a, b, c
∈
N
with
a
and
b
coprime. Show that if
a

c
and
b

c
then
ab

c
.
Solution
As
a, b
are coprime, there exist
m, n
∈
Z
such that 1 =
ma
+
nb
and
therefore
c
=
mac
+
nbc
. As
a
and
b
divide
c
, there exist
x
and
y
in
N
such
that
c
=
xa
and
c
=
yb
, whence subtitution yields
c
=
mayb
+
nbxa
= (
my
+
nx
)
ab.
As
Z
is closed under addition and multiplication, this means that
ab
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Unformatted text preview: divides c as required. Here is an alternative (briefer and neater) argument. As above, c = yb for y ∈ N . Now, a divides c = yb and is coprime to b , so a  y ; thus, y = n a for n ∈ N . It follows that c = yb = n ab so that c is a multiple of ab as required. 1...
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This note was uploaded on 11/12/2011 for the course MAS 3300 taught by Professor Staff during the Fall '08 term at University of Florida.
 Fall '08
 Staff

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