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.