The difference between a 2-digit number and the number obtained by interchanging the positions of the digits is 54. Consider the following statements : 1. The sum of the two digits of the number can be determined only if the product of the two digits is known. 2. The difference between the two digits of the number can be determined. Which of the above statements is/are correct ?
- (a) 1 only
- (b) 2 only ✓ UPSC's answer
- (c) Both 1 and 2
- (d) Neither 1 nor 2
Why the answer is (b)
• Let the number be 10a + b. Then (10a + b) − (10b + a) = 9(a − b) = 54, so the digits differ by 6.
• Statement 2 is correct: the difference between the digits is always 6.
• The possible numbers are 60, 71, 82 and 93 (or their reverses), with digit sums 6, 8, 10 and 12. The sum is not fixed, but it can be found in several ways — for example from either digit, or from the product (0, 7, 16, 27). So it is not true that the sum can be found 'only if' the product is known; statement 1 is not correct.
• Hence 2 only, option (b).
Why the other options are wrong
- (a) 1 only
- Statement 1 is not correct (the sum can be found in other ways, e.g. from one digit), while statement 2 is correct.
- (c) Both 1 and 2
- Statement 1 is not correct: knowing the product is not the only way to find the sum.
- (d) Neither 1 nor 2
- Statement 2 is correct: 9(a − b) = 54 gives a digit difference of 6.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2021, held on 10 October 2021. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.