MATH 74 HOMEWORK 3: DUE MONDAY 2/12
1.
If
a
and
b
are natural numbers, and
b > a
, and
a

b
, then there are
q
∈
N
and
0
< r < a
satisfying
b
=
qa
+
r.
In class we proved gcd(
a, b
) = gcd(
a, r
) by iterating a subtration rule gcd(
a, b
) =
gcd(
a, b

a
) repeatedly. In this exercise you will give a different, more direct proof.
1(a).
Let
a, b, q, r
be as above. Suppose you have a natural number
x
that is in
D
(
a, b
). Show, making explicit references to the definitions of the relevant things,
that
x
is also an element of
D
(
a, r
).
1(b).
Give a similarly explicit proof that if
x
is in
D
(
a, r
) then
x
is in
D
(
a, b
).
Remark.
The
D
(
,
)
notation is explained in the lecture outlines, if you missed
it. Together, the statements 1(a) and 1(b) show that
D
(
a, r
) =
D
(
a, b
)
, and then
gcd(
a, r
) = gcd(
a, b
)
follows from the definition of
gcd
.
Remark.
In the previous exercise, you showed that two pairs of numbers had the
same
gcd
by showing that they had the same set of common divisors. This raises
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 Fall '07
 COURTNEY
 Natural Numbers, Natural number, Prime number, Greatest common divisor

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