What is the two-digit integer N? Statement 1: The sum of the digits of N is 11. Statement 2: When the digits of N are reversed, the new number is 27 more than N.

ATQ, Let N = 10a + b (a: tens digit, b: units digit). From Statement 1: a + b = 11. Many possibilities (29, 38, 47, 56, 65, 74, 83, 92...). Not sufficient alone. From Statement 2: Reversed number = 10b + a. Given 10b + a = (10a + b) + 27 => 10b + a = 10a + b + 27 => 9b − 9a = 27 => b − a = 3. Again multiple possibilities; not sufficient alone. Combining: a + b = 11 b − a = 3 Add the equations: 2b = 14 ⇒ b = 7. Then a = 11 − 7 = 4. So N = 47, uniquely. Thus both statements together are sufficient, neither alone.