A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question : What are the values of m and n, where m and n are natural numbers? Statement-I : m + n > mn and m > n. Statement-II : The product of m and n is 24. Which one of the following is correct in respect of the above Question and the Statements?
- (a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
- (b) The Question can be answered by using either Statement alone
- (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone ✓ UPSC's answer
- (d) The Question cannot be answered even by using both the Statements together
Why the answer is (c)
• Statement I (m + n > mn and m > n) allows for multiple pairs of natural numbers, such as (3, 2) and (4, 3), so it is insufficient alone.
• Statement II (mn = 24) allows for multiple factor pairs like (24, 1), (12, 2), (8, 3), and (6, 4), so it is insufficient alone.
• Combining both statements, we test the factor pairs of 24 against the condition m + n > mn.
• For (24, 1): 25 > 24 (True), but m > n is True. However, let's check other pairs.
• For (12, 2): 14 > 24 (False).
• For (8, 3): 11 > 24 (False).
• For (6, 4): 10 > 24 (False).
• The only pair satisfying m + n > mn is (24, 1) where 24+1=25 > 24*1=24, and m=24 > n=1. Thus, m=24 and n=1 is the unique solution.
Why the other options are wrong
- (a) The Question can be answered by using one of the Statements alone, but cannot be answered…
- Statement I alone yields multiple valid pairs like (3,2) and (4,3), so it cannot answer the question alone.
- (b) The Question can be answered by using either Statement alone
- Neither statement alone provides a unique pair of values for m and n.
- (d) The Question cannot be answered even by using both the Statements together
- Using both statements together uniquely identifies m=24 and n=1, so the question can be answered.
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.