Question

Which of the following statement is true?

• A. There does not exist an integer b such that for $a>1$, $a\times b=b\times a=b$
• B. The product of a positive and a negative integer is positive.
• C. Subtraction follows commutative property.
• D. All are false.

Answer 1

The correct option is A There does not exist an integer b such that for $a>1$, $a\times b=b\times a=b$

Let's look at the statements one by one.

Statement 1: There does not exist an integer b such that for $a>1$, $a\times b=b\times a=b$.
It is true.
Breaking down the equation as:
$a\times b=b$
$b\times a=b$
If we take any value for $a>1$ say 2, then
$2\times b\ne b$ and $b\times 2\ne b$
Hence, $a\times b\ne b\times a\ne b$
$\therefore$ There does not exist an integer b such that for $a>1$, $a\times b=b\times a=b$.

Statement 2: The product of a positive and a negative integer is positive.
It is false because the product of a positive and a negative integer is negative not positive.

Statement 3: Subtraction follows commutative property.
This is also false because commutative law of integers under subtraction states that $a-b\ne b-a.$