Math 310, Section 2, Fall 2009
Remarks on HW Section 1.1
From Section 1.1:
1a,b,c, 6, 8
1 Find the quotient and remainder when
a
is divided by
b
:
(a)
a
= 302,
b
= 19
(b)
a
=

302,
b
= 19
(c)
a
= 0,
b
= 19
Solution:
(a) 302 = 19
·
15 + 17,
q
= 15, 0
≤
r
= 17
< b
= 19.
(b)

302 = 19
·
16 + 2,
q
=

16, 0
≤
r
= 2
< b
= 19.
(c) 0 = 19
·
0 + 0,
q
= 0, 0
≤
r
= 0
< b
= 19.
6 Use the Division Algorithm to prove that every odd integer is either of
the form 4
k
+ 1 or of the form 4
k
+ 3 for some integer
k
.
Proof:
An even integer is, by deﬁnition, an integer divisible by 2, that is,
an integer that has a remainder of 0 after division by 2. An odd integer is,
then, an integer not divisible by 2, that is, an integer that has a remainder
of 1 after division by 2.
Let
a
be an odd integer. Apply the Division Algorithm to
a
and
b
= 4:
there exist unique integers
q
and
r
such that
a
= 4
q
+
r
where 0
≤
r <
4.
That is,