Use a direct proof to show that the product of two odd numbers is odd.

Question

Rosemary McBride · Accepted Answer

n*m is odd if n and m are both odd. n and m are actual numbers.
n=2a+1 Odd integer = 2k+1
m=2b+1 Multiply terms
n*m=(2a+1)(2b+1)
n*m=4ab+2a+2b+1 Factor out 2
n*m=2(2ab+a+b)+1
n*m=2k+1 k=2ab+a+b=integer
$∴ n \cdot m \to o d d$
Result:
$n \cdot m \to o d d$

RizerMix · Accepted Answer

To prove that the product of two odd numbers is odd, we can use a direct proof.
Let's assume that we have two odd numbers,

a

and

b

. By definition, an odd number can be represented as

2 n + 1

, where

n

is an integer.
Therefore, we can express

a

and

b

as:

a = 2 m + 1

and

b = 2 k + 1

,
where

m

and

k

are integers.
Now, let's calculate the product of

a

and

b

:

a b = (2 m + 1) (2 k + 1)

.
To simplify this expression, we can expand it using the distributive property:

a b = 4 m k + 2 m + 2 k + 1

.
Notice that

4 m k + 2 m + 2 k

is an even number since it can be expressed as

2 (2 m k + m + k)

.
Therefore, we can rewrite the expression as:

a b = 2 (2 m k + m + k) + 1

.
Since

2 m k + m + k

is an integer, let's call it

p

:

a b = 2 p + 1

.
We can see that the product of

a

and

b

can be expressed as an odd number

2 p + 1

.
Hence, we have proved that the product of two odd numbers is odd using a direct proof.

Vasquez · Accepted Answer

Step 1:Therefore, we can express a and b as a=2n1+1 and b=2n2+1, respectively, where n1 and n2 are integers.The product of a and b is given by ab=(2n1+1)(2n2+1).Expanding the expression, we get:ab=4n1n2+2n1+2n2+1Step 2:Now, let&#039;s factor out 2 from the first three terms:ab=2(2n1n2+n1+n2)+1Since 2n1n2+n1+n2 is an integer (sum of integers is an integer), we can rewrite the expression as:ab=2k+1where k=2n1n2+n1+n2 is also an integer.Hence, we have expressed ab in the form 2k+1, where k is an integer. This shows that the product of two odd numbers is odd.Thus, the statement has been proven using a direct proof.

Use a direct proof to show that the product of two odd numbers is odd.

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