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

Correct Option (3)

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

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

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

Simplifying the equation:

$$9x - 9y = 54$$9x−9y=54

$$9(x - y) = 54$$9(x−y)=54

$$x - y = 6$$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 ($x + y$x+y) can be determined only if the product of the two digits ($xy$xy) is known.

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

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

$$(x + y)² = (6)² + 4xy$$(x+y)2=(6)2+4xy

$$(x + y)² = 36 + 4xy$$(x+y)2=36+4xy

From this relationship, it is evident that if the product $xy$xy is known, then $(x + y)²$(x+y)2 can be calculated, and consequently, $x + y$x+y (the sum of the digits) can be uniquely determined (as $x$x and $y$y are positive digits, their sum must be positive). Without knowing $xy$xy, $x+y$x+y cannot be uniquely determined (e.g., if $x-y=6$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.