The difference between a 2-digit number and the number obtained by interchanging the positions of the digits is 54. Consider the following statements: 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?

Let the 2-digit number be represented as 10x + y, where x is the tens digit and y is the units digit. The number obtained by interchanging the positions of the digits is 10y + x.

According to the problem, the difference between these two numbers is 54. (10x + y) - (10y + x) = 54 10x - x + y - 10y = 54 9x - 9y = 54 9(x - y) = 54 x - y = 54 / 9 x - y = 6

Now let's analyze the given statements:

Statement 1: "The sum of the two digits of the number can be determined only if the product of the two digits is known." We know that x - y = 6. Let's list the possible pairs of digits (x, y) that satisfy this condition, keeping in mind that x is a digit from 1-9 and y is a digit from 0-9:

1. If y = 0, then x = 6. The number is 60. Sum (x+y) = 6. Product (x*y) = 0.
2. If y = 1, then x = 7. The number is 71. Sum (x+y) = 8. Product (x*y) = 7.
3. If y = 2, then x = 8. The number is 82. Sum (x+y) = 10. Product (x*y) = 16.
4. If y = 3, then x = 9. The number is 93. Sum (x+y) = 12. Product (x*y) = 27. Since the sum (x+y) can be 6, 8, 10, or 12, it is not uniquely determined from the given information (x-y=6). The statement claims the sum can be determined only if the product is known. This implies that knowing the product is the sole way to determine the sum. However, if we were given other information, such as the value of one of the digits (e.g., if we knew y=1, then x=7, and x+y=8), we could determine the sum without knowing the product. Since there are other ways to determine the sum, the "only if" condition makes Statement 1 incorrect.

Statement 2: "The difference between the two digits of the number can be determined." From our calculation above, we found that x - y = 6. This value is uniquely determined from the given information. Therefore, Statement 2 is correct.

Based on the analysis, only Statement 2 is correct.

The final answer is B