Correct Option (3)
The prime numbers less than 10 are 2, 3, 5, and 7.
Given that x, y, and z are distinct prime numbers each less than 10, we consider all possible combinations of three distinct primes from this set:
- Case 1: x = 2, y = 3, z = 5. The sum S = 2 + 3 + 5 = 10. The unit digit of S is 0.
- Case 2: x = 2, y = 3, z = 7. The sum S = 2 + 3 + 7 = 12. The unit digit of S is 2.
- Case 3: x = 2, y = 5, z = 7. The sum S = 2 + 5 + 7 = 14. The unit digit of S is 4.
- Case 4: x = 3, y = 5, z = 7. The sum S = 3 + 5 + 7 = 15. The unit digit of S is 5.
The possible unit digits for S are {0, 2, 4, 5}.
Evaluating the given statements:
- Statement 1: "The unit digit of S can be 0." This is correct, as demonstrated by the sum 2 + 3 + 5 = 10.
- Statement 3: "The unit digit of S can be 5." This is correct, as demonstrated by the sum 3 + 5 + 7 = 15.
Therefore, statements 1 and 3 are correct.
Incorrect Options:
- Statement 2: "The unit digit of S can be 9." This is incorrect. Based on the possible sums calculated (10, 12, 14, 15), the unit digits are {0, 2, 4, 5}. The digit 9 is not among these possible unit digits.