Question

Statement (A): The sum of even number of times of an odd number is even Statement (B): The sum of odd number of times of an odd number is even. a) Both A and B are true b) Both A and B are false c) A is true and B is false d) A is false and B is true​

Answer 1

$\large\text{\underline{Let's begin:-}}$

Statement A: The sum of an even number of times of an odd number is even.

Statement B: The sum of an odd number of times of an odd number is even.

Adding a number repeatedly can be replaced by a product.

Statement A is equivalent to $\text{(even)}\times\text{(odd)}=\text{(even)}$.

Statement B is equivalent to $\text{(odd)}\times\text{(odd)}=\text{(odd)}$.

An odd number and an even number can be shown as $2n-1$ and $2n$ respectively.

$\large\text{\underline{Solution:-}}$

Statement A:

In statement A, two numbers will be $2a$ and $2b-1$.

(To prove: $\text{(even)}\times\text{(odd)}=\text{(even)}$)

Finding the product of two numbers,

$\hookrightarrow 2a\times(2b-1)=4ab-2a$

$\hookrightarrow 2a\times(2b-1)=2(2ab-a)$

Hence the product is an even number. (Correct.)

Statement B:

In statement B, two numbers will be $2a-1$ and $2b-1$.

(To prove: $\text{(odd)}\times\text{(odd)}=\text{(odd)}$)

Finding the product of two numbers,

$\hookrightarrow (2a-1)(2b-1)=4ab-2a-2b+1$

$\hookrightarrow (2a-1)(2b-1)=2(2ab-a-b)+1$

Hence the product is an odd number. (Correct.)

$\large\text{\underline{Result:-}}$

Both statements are true, so option (a) is the correct answer.

Answer 2

G I V E N :

Statement (A): The sum of eve number of times of an odd number is even.

Statement (B): The sum of odd number of times of an odd number is even.

a) Both A and B are true

b) Both A and B are false

c) A is true and B is false

d) A is false and B is true

S O L U T I O N :

STATEMENT (A)

According to the question, we are given that if an even number is multiplied with an odd number the result is even

Let us understand with an easy example :

Let us assume the even number be 2

The odd number be 3

So, now as per the condition if we multiply the even number with an odd number we will get an even number

Odd × Even = Even

3 × 2 = 6

So as we know that 6 is an even number so statement A is correct

STATEMENT (B)

Now, statement B states that odd is multiplied by another odd number so the result will be odd

Let us assume the first odd number be 9

The second odd one be 17

Now, getting their products

Odd × Odd = Odd

9 × 17 = 153

So, 153 is an odd number hence statement B is also correct

As both statements are correct

• Option A Is The Correct Answer

[Note : If we add an odd number with another odd no. the result will be an even number whereas if we add an odd one with an even no. the result will be an odd number. Incase of an even number if we both the even numbers we will get an even number]