MATH 74 HOMEWORK 3 SOLUTIONS
1.
We are given that
x
∈
D
(
a, b
). This means (by definition of
D
(
a, b
)) that
x

a
and
x

b
, and this means (by definition of “divides”) that there are
k
∈
N
and
l
∈
N
with
a
=
kx
and
b
=
lx
.
We are supposed to show that
x
∈
D
(
a, r
), ie, that
x

a
and
x

r
. As we already
know that
x

a
, it is thus enough to show that
x

r
. But
r
=
b

qa
=
lx

qkx
= (
l

qk
)
x.
Note that
l

qk
is certainly an integer, and it has to be positive since
r
and
x
both
are, so it is a natural number, and hence
x

r
.
1(b).
If
x
is in
D
(
a, r
) then (by definition of
D
(
a, r
)) we have that
x

a
and
x

r
.
We are supposed to show that
x
must also be in
D
(
a, b
), ie, that
x

a
and
x

b
.
Since we know
x

a
already, it is thus enough to show that
x

b
.
To do this, note that as
x

a
there is
k
∈
N
satisfying
a
=
kx
, and as
x

r
there
is
l
∈
N
satisfying
r
=
lx
. We then see that
b
=
qa
+
r
=
qkx
+
lx
= (
qk
+
l
)
x.
Since
q, k, l
are natural numbers, so is
qk
+
l
. This verifies that
x
divides
b
.
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 Fall '07
 COURTNEY
 Math, Prime number, Greatest common divisor, 2k, Euclidean algorithm, gcd

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