Let pp, qq and rr be 2-digit numbers where p < q < r. If pp + qq + rr = tt0, where tt0 is a 3-digit number ending with zero, consider the following statements : 1. The number of possible values of p is 5. 2. The number of possible values of q is 6. Which of the above statements is/are correct ?

(a) 1 only
(b) 2 only
(c) Both 1 and 2 ✓ UPSC's answer
(d) Neither 1 nor 2

Why the answer is (c)

• pp, qq, rr are repdigits, so pp=11p, qq=11q, rr=11r, and tt0=110t.

• Equation gives 11(p+q+r)=110t, hence p+q+r=10t.

• Since p,q,r are distinct digits 1 to 9 with p<q<r, their sum can only be 10 or 20.

• For sum 10, triples are (1,2,7), (1,3,6), (1,4,5), (2,3,5), giving p values {1,2} and q values {2,3,4}.

• For sum 20, triples are (3,8,9), (4,7,9), (5,6,9), (5,7,8), giving p values {3,4,5} and q values {6,7,8}.

• Union gives p values {1,2,3,4,5} (5 values) and q values {2,3,4,6,7,8} (6 values), so both statements are correct, option (c).

Why the other options are wrong

(a) 1 only: Option (a) is wrong because q also has six possible values, so statement 2 is correct as well.
(b) 2 only: Option (b) is wrong because p has five possible values, so statement 1 is correct as well.
(d) Neither 1 nor 2: Option (d) is wrong because both p and q have the stated counts, making both statements correct.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.