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?

(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only ✓ UPSC's answer
(d) 1, 2 and 3

Why the answer is (c)

• The distinct prime numbers less than 10 are 2, 3, 5, and 7.

• To check Statement 1 (unit digit 0), consider the set {2, 3, 5}; their sum is 10, which has a unit digit of 0, making this statement correct.

• To check Statement 2 (unit digit 9), the possible sums of any three distinct primes from {2, 3, 5, 7} are 10 (2+3+5), 12 (2+3+7), 14 (2+5+7), and 15 (3+5+7); none of these sums have a unit digit of 9, making this statement incorrect.

• To check Statement 3 (unit digit 5), consider the set {3, 5, 7}; their sum is 15, which has a unit digit of 5, making this statement correct.

• Since Statements 1 and 3 are correct and Statement 2 is incorrect, the correct option is (c).

Why the other options are wrong

(a) 1 and 2 only: Statement 2 is incorrect because no combination of three distinct primes less than 10 sums to a number with a unit digit of 9.
(b) 2 and 3 only: Statement 2 is incorrect as no valid sum yields a unit digit of 9, and Statement 1 is actually correct.
(d) 1, 2 and 3: Statement 2 is incorrect because the sum of any three distinct primes less than 10 cannot have a unit digit of 9.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.