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A4 US

Consider the statement:

If $a b$ is an even number, then $a$ or $b$ is even.
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Which of the proofs below appear to be valid proofs of this statement? Note: You can assume all the algebra below is correct (because it is).

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1.

Suppose

a

and

b

are odd. That is,

a = 2 k + 1

and

b = 2 m + 1

for some integers

k

and

m .

Then

\begin{aligned} a b & = (2 k + 1) (2 m + 1) \\ = 4 k m + 2 k + 2 m + 1 \\ = 2 (2 k m + k + m) + 1 . \end{aligned}

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Therefore

a b

is odd.

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2.

Assume that

a

b

is even -- say it is

a

(the case where

b

is even will be identical). That is,

a = 2 k

for some integer

k .

Then

\begin{aligned} a b & = (2 k) b \\ = 2 (k b) . \end{aligned}

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Thus

a b

is even.

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3.

Suppose that

a b

is even but

a

and

b

are both odd. Namely,

a b = 2 n,

a = 2 k + 1

and

b = 2 j + 1

for some integers

n,

k,

and

j .

Then

\begin{aligned} 2 n & = (2 k + 1) (2 j + 1) \\ 2 n & = 4 k j + 2 k + 2 j + 1 \\ n & = 2 k j + k + j + \frac{1}{2} . \end{aligned}

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But since

2 k j + k + j

is an integer, this says that the integer

n

is equal to a non-integer, which is impossible.

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4.

Let

a b

be an even number, say

a b = 2 n,

and

a

be an odd number, say

a = 2 k + 1 .

\begin{aligned} a b & = (2 k + 1) b \\ 2 n & = 2 k b + b \\ 2 n - 2 k b & = b \\ 2 (n - k b) & = b . \end{aligned}

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Therefore

b

must be even.

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5.

Which of the proofs above are valid proofs of the statement?

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Proof 1.
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Proof 2.
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This is a valid proof, but not of the statement given. It is proving something else.
Proof 3.
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Proof 4.
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