# Q. 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

(d) Neither 1 nor 2

Correct Answer: (c) Both 1 and 2

#### Question from UPSC Prelims 2023 CSAT

**Explanation : **

## Analysis of 2-Digit Numbers – pp, qq & rr

The problem is based on the concept of number theory and algebra.

The given condition is that pp, qq and rr are 2-digit numbers where p< q< r and their sum is a 3-digit number ending with zero.

This means that the sum of the digits p, q and r is either 10 or 20, as the sum of their double-digit representations (pp, qq, rr) is a three-digit number ending in zero (either 110 or 220).

For the first statement, we need to find the possible values of p. Since p is the smallest digit and the sum of p, q and r is either 10 or 20, the possible values of p can be 1 or 2 for the sum 10, and 1, 2, 3, 4 or 5 for the sum 20. So, there are 5 possible values for p, making the first statement correct.

For the second statement, we need to find the possible values of q. Since q is the middle digit and the sum of p, q and r is either 10 or 20, the possible values of q can be 2, 3 or 4 for the sum 10, and 6, 7 or 8 for the sum 20. So, there are 6 possible values for q, making the second statement correct.

Therefore, both statements are correct, and the answer is (c) Both 1 and 2.