Question

If a product is even what does it say about the factors?. which statement is true?

a)Both factors can be odd
b)one factor can be odd
c)Both Factors have to be even​

Answer 1

Answer:

b) one factor can be odd

Explanation:

If a number is a product of two numbers, then the following three cases are possible:

The number is a product of

• One odd and One even number
• Two even numbers
• Two odd numbers

let us examine all three cases one by one:

Product of one odd and one even number.

An odd number is in the form = 2n + 1

An even number is in the form = 2n

Their product:

⟹ 2n( 2n + 1 )

⟹ 4n² + 2n

⟹ 2( 2n² + n )

We can see, their product can be return the form 2 × k, this is shows it is a multiple of 2, thus product of one odd and one even number is an even number.

Product of two even numbers.

Let first even number = 2n

Let the second even number = 2m

Their product:

⟹ 2n × 2m

⟹ 4mn

⟹ 2( 2mn )

We can see, their product can be return the form 2 × k, this is shows it is a multiple of 2, thus product of two even number is an even number.

Product of two odd numbers:

let the first odd number = 2n + 1

let the second odd number = 2m + 1

Their product:

⟹ 2n + 1 ( 2m + 1 )

⟹ 2n( 2m + 1 ) + 1( 2m + 1 )

⟹ 4mn + 2n + 2m + 1

⟹ 2( mn + n + m ) + 1

We can see, 2( mn + n + m ) is an even number as it is in the form 2 × k, but there is a "1" added to an even number. So, the product of two odd number is an odd number.

We've seen that for the product of two number to be even, the both 2 number does not have to be even. Even if the one number is odd, then the product can result to an even number too.

So, option b) one factor can be odd is the correct option.