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MAE3811–Sp. 2010 Mahoney–1
From Last time…. Division In Depth
Recall, it was recent, that for any integers
a
and
b
, with
b not equal to 0
,
a ÷ b = c
if
and
only if
c
is the
unique
whole number such that
b∙c = a
.
We say
b
divides
a
, written
b  a
, if, and only if, there is a unique integer
q
such
that
a = bq
.
If
b  a
, then
b
is a ______________ or a __________________ of
a
.
a
is a ___________________ of
b
.
q
usually stands for the ___________________ of
a
and
b
True or False
(a  b)  a
2
–b
2
, a not equal to b
0  0
MAE3811–Sp. 2010 Mahoney–2
Properties of divisibility
Recall a  b means….
Theorem: a  b if and only if a  b
Proof:
Theorem: if d  a, and n is any integer, then d  na
Proof:
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MAE3811–Sp. 2010 Mahoney–3
Properties of divisibility
II
Let a, b, and d be any integers with d ≠ 0.
Theorem: If d  a and d  b,
then d  (a + b).
Proof:
Corollary:
d  a and d  b,
then d  (a  b)
Proof:
MAE3811–Sp. 2010 Mahoney–4
Proof by Contradiction
Proof by contradiction is a method of Proof often used when you wish to prove the
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 Spring '11
 Mahoney

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