Consider the following statements in respect of the sum S = x + y + z, where x, y and z are distinct prime numbers each less than 10:
1. The unit digit of S can be 0.
2. The unit digit of S can be 9.
3. The unit digit of S can be 5.

Which of the statements given above are correct?

Correct Option: (c) 1 and 3 only

We are given S = x + y + z, where x, y, and z are distinct prime numbers less than 10. The prime numbers less than 10 are: 2, 3, 5, and 7. Since x, y, and z are distinct, we must choose any 3 prime numbers from this set of 4. There are exactly 4 combinations: 1) {2, 3, 5}: S = 2 + 3 + 5 = 10 (unit digit is 0); 2) {2, 3, 7}: S = 2 + 3 + 7 = 12 (unit digit is 2); 3) {2, 5, 7}: S = 2 + 5 + 7 = 14 (unit digit is 4); 4) {3, 5, 7}: S = 3 + 5 + 7 = 15 (unit digit is 5). The possible values of S are 10, 12, 14, and 15, and the possible unit digits are 0, 2, 4, and 5. Let us analyze the given statements: Statement 1 says the unit digit can be 0, which is correct (from S = 10). Statement 2 says the unit digit can be 9, which is incorrect as 9 is not in our set of possible unit digits. Statement 3 says the unit digit can be 5, which is correct (from S = 15). Thus, only Statements 1 and 3 are correct. The correct option is (c).