# Q. Let X be a two-digit number and Y be another two-digit number formed by interchanging the digits of X. If (X+Y) is the greatest two-digit number, then what is the number of possible values of X?

a) 2

b) 4

c) 6

d) 8

Correct answer: d) 8

##### Question from UPSC Prelims 2024 CSAT Paper

**Explanation:**

## X+Y is the greatest two-digit number

Let’s approach this step-by-step:

1) First, we need to understand what the question is asking:

– X is a two-digit number

– Y is formed by interchanging the digits of X

– The sum of X and Y is the greatest two-digit number

2) The greatest two-digit number is 99.

3) So, we’re looking for pairs of numbers X and Y that add up to 99.

4) Let’s represent X as ‘ab’ where ‘a’ is the tens digit and ‘b’ is the ones digit.

Then Y would be represented as ‘ba’.

5) We can write an equation:

ab + ba = 99

6) This can be rewritten as:

(10a + b) + (10b + a) = 99

11a + 11b = 99

a + b = 9

7) Now, we need to find all possible integer combinations of a and b where:

– a and b are single digits (0-9)

– a + b = 9

8) The possible combinations are:

0 and 9

1 and 8

2 and 7

3 and 6

4 and 5

5 and 4

6 and 3

7 and 2

8 and 1

9 and 0

9) However, we can’t use 0 as the first digit of a two-digit number, so we eliminate the first and last combinations.

10) This leaves us with 8 possible combinations.

Therefore, the correct answer is d) 8. There are 8 possible values for X.