The number 1102 in base 3 is equivalent to 123 in which base system? ISRO 2013

First, convert the base 3 number 1102 to decimal: (1 * 3^3) + (1 * 3^2) + (0 * 3^1) + (2 * 3^0) = 27 + 9 + 0 + 2 = 38. Next, let the unknown base be x. The number 123 in base x equals (1 * x^2) + (2 * x) + 3. Equate this to the decimal value: x^2 + 2x + 3 = 38. Simplify to x^2 + 2x - 35 = 0. Factor the quadratic equation: (x + 7)(x - 5) = 0. The possible solutions are x = -7 or x = 5. Since a base must be positive, the answer is 5.