MATH 135 Assignment 1
This assignment is due at 8:30am on Wednesday January 13, in the drop boxes opposite the
Tutorial Centre, MC 4067.
1. Let
a, b, c, d
be integers. Suppose that
c

a
and
d

b
. Prove that
cd

ab
.
Solution:
Since
c

a
there exists an integer
q
such that
a
=
cq
. Since
d

b
there exists an
integer
r
such that
b
=
dr
. Therefore
ab
=
cd
(
rq
), so
cd

ab
by deﬁnition (because
qr
is an
integer).
2. Let
a
and
b
be integers. Prove that
gcd
(
a, b
)

gcd
(
a
+
b, a

b
)
.
Solution:
Let
d
=
gcd
(
a
+
b, a

b
). Then we know there exist integers
x
and
y
such that
(
a
+
b
)
x
+ (
a

b
)
y
=
d.
Therefore
a
(
x
+
y
) +
b
(
x

y
) =
d.
By deﬁnition
gcd
(
a, b
)

a
and
gcd
(
a, b
)

b
. Therefore
gcd
(
a, b
)

d
, as required.
3. Let
a, b, c
be integers. Suppose that
gcd
(
a, b
) = 1 and
c

(
a
+
b
). Prove that
gcd
(
a, c
) = 1.
Solution:
Let
d
=
gcd
(
a, c
). Then by deﬁnition
d

a
and
d

c
, and
d
≥
0.
Since
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 Spring '08
 ANDREWCHILDS
 Integers, Greatest common divisor, Euclidean algorithm, gcd, Tutorial Centre

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