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

Correct Option (3)

Let the two-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 will be $10y+x$.

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

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

Simplifying the equation:

$$9x-9y=54$$

$$9(x-y)=54$$

$$x-y=6$$

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 ($xy$) is known.

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

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

$$(x+y)²=(6)²+4xy$$

$$(x+y)²=36+4xy$$

From this relationship, it is evident that if the product $xy$ 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 $xy$, $x+y$ 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.