Answer
See the explanation
Work Step by Step
A Turing machine can be designed to determine whether a given number is prime by following these steps:
1. **Input Handling**: The Turing machine takes the given number as input on its tape.
2. **Initialization**: The machine initializes its state and sets up its tape to represent the number and other necessary data structures.
3. **Divisibility Testing**: The machine iteratively tests whether the input number is divisible by any integer from 2 to the square root of the input number. It does this by simulating division operations and checking for remainders.
4. **Decision Making**: Based on the divisibility testing results, the machine decides whether the input number is prime or composite.
5. **Output**: Finally, the Turing machine halts and outputs either "prime" or "composite" based on its decision.
This process demonstrates how a Turing machine can systematically analyze the divisibility properties of a given number to determine its primality.
