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 number be xy, i.e. 10x + y The number obtained on interchanging the positions of the digits will be yx, i.e. 10y + x Difference between these two numbers = (10x + y) – (10y + x) = 9x – 9y = 9 (x – y) = 54 So, 9 (x – y) = 54 Or x – y = 6 So, statement 2 is correct. The possible pair of such two-digit numbers are: (17, 71), (28, 82), (39, 93) Respective product of their digits are: 1 × 7 = 7; 2 × 8 = 16; 3 × 9 = 27 So, we can determine the exact number-pair and hence the sum of their digits, if we know another relationship between the two numbers i.e. a second equation. This can either be product of their digits or Ratio of the two digits, ie. x/y. For example, if we know x/y is 1/7 we know that the sum would be 8 So, statement 1 is not correct.