The only possible number base for 235\(_x\) among the following is ?

In any positional number system with base \(x\), every digit used in a number must be strictly less than the base itself. A base \(x\) system only has the digits \(0\) through \(x-1\) available, so a digit equal to or greater than the base can never legally appear.

The number \(235_x\) contains the digit \(5\), which means the base must satisfy \(x > 5\), or equivalently \(x \geq 6\). Testing the possible bases given: base \(3\) only allows digits \(0\), \(1\), and \(2\); base \(4\) only allows digits \(0\) to \(3\); and base \(5\) only allows digits \(0\) to \(4\). None of these can contain a digit \(5\), so all three are invalid. Base \(6\) allows digits \(0\) through \(5\), which comfortably accommodates every digit in \(235_x\), making it the only valid choice.

Whenever a number is written with an unknown base and contains a specific digit, the base must always be greater than the largest digit present; use this check first to eliminate impossible bases quickly.