Correct Option (3)

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

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

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

Simplifying the equation:

$9 x - 9 y = 54$

$9 (x - y) = 54$

$x - y = 6$

This equation directly determines the difference between the two digits of the number. Therefore, statement 2 is correct.

For statement 1, we consider whether the sum of the two digits ( $x + y$ ) can be determined only if the product of the two digits ( $x y$ ) is known.

We utilize the algebraic identity: $(x + y) ² = (x - y) ² + 4 x y$ .

Since we have established that $x - y = 6$ , we can substitute this value into the identity:

$(x + y) ² = (6) ² + 4 x y$

$(x + y) ² = 36 + 4 x y$

From this relationship, it is evident that if the product $x y$ is known, then $(x + y) ²$ can be calculated, and consequently, $x + y$ (the sum of the digits) can be uniquely determined (as $x$ and $y$ are positive digits, their sum must be positive). Without knowing $x y$ , $x + y$ cannot be uniquely determined (e.g., if $x - y = 6$ , possible pairs are (7,1), (8,2), (9,3) with sums 8, 10, 12 respectively).

Therefore, statement 1 is also correct.

Incorrect Options:

Options 1, 2, and 4 are incorrect because both statement 1 and statement 2 are correct, as demonstrated by the derivations.