The difference between a 2-digit number and the number obtained by interchanging the positions of the digits is 54.

1. The sum of the two digits of the number can be determined only if the product of the two digits is known.
2. The difference between the two digits of the number can be determined.

Which of the above statements is/are correct?

Correct Answer: The correct option is 2 only.

Step-by-Step Explanation:

Let us analyze the problem by representing the 2-digit number algebraically.

Let the tens digit of the 2-digit number be $x$ and the units digit be $y$.

The original number can be written as:
$10x+y$

When the positions of the digits are interchanged, the tens digit becomes $y$ and the units digit becomes $x$. Therefore, the new number is:
$10y+x$

According to the given condition, the difference between the original number and the number obtained by interchanging its digits is 54:

$(10x+y)-(10y+x)=54$

Simplifying the left-hand side of the equation:

$10x-x+y-10y=54$

$9x-9y=54$

Dividing both sides by 9:

$x-y=6$

Now, let us evaluate the two given statements based on this result:

Statement 2: The difference between the two digits of the number can be determined.
As derived above, the difference between the two digits $x-y=6$. This is a definite value. Thus, Statement 2 is correct.

Statement 1: The sum of the two digits of the number can be determined only if the product of the two digits is known.
Using algebraic identities, we know that:
${(x+y)}^2={(x-y)}^2+4xy$

Since $x-y=6$, we have:
${(x+y)}^2=36+4xy$

To find the sum of the digits $(x+y)$, knowing the product $xy$ alone is not sufficient because multiple pairs of digits $(x,y)$ can satisfy $x-y=6$ (for instance, $(9,3)$, $(8,2)$, or $(7,1)$). However, the statement asserts that it can be determined only if the product is known, which is also incorrect because knowing one of the digits or the number itself could also yield the sum. Therefore, Statement 1 is not correct.

Hence, only Statement 2 is correct.