Which of the following positive integers is a prime number?

Step 1 (Definition of Prime): A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself.

Step 2 (Verify candidates):

• For 105: The last digit is 5, so it is divisible by 5 ($105=5\times 21$). Not prime.

• For 111: The sum of the digits is $1+1+1=3$, so it is divisible by 3 ($111=3\times 37$). Not prime.

• For 115: The last digit is 5, so it is divisible by 5 ($115=5\times 23$). Not prime.

• For 101: The prime numbers smaller than $\sqrt{101}\approx 10$ are 2, 3, 5, and 7. Testing divisibility:

• 101 is odd (not divisible by 2).

• Sum of digits is 2 (not divisible by 3).

• Last digit is 1 (not divisible by 5).

• $101=7\times 14+3$ (not divisible by 7).

Since 101 is not divisible by any prime up to its square root, it is a prime number.

Step 3 (Conclusion): The prime number is 101, corresponding to option index 0.