Q: 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
d) The Question cannot be answered even by using both the Statements together
Correct answer: c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
Question from UPSC Prelims 2024 CSAT
Explanation :
Statement I Alone:
1. Rearranging m + n > mn gives (m-1)(n-1) < 1
2. Since m and n are natural numbers, and m > n:
– n must equal 1
– m can be any natural number > 1
This alone doesn’t give unique values.
Statement II Alone:
1. Factors of 24 give these pairs:
(1,24), (2,12), (3,8), (4,6), (6,4), (8,3), (12,2), (24,1)
This alone doesn’t give unique values.
Both Statements Together:
1. From Statement I, n must be 1
2. From Statement II, mn = 24
3. Substituting n = 1:
m × 1 = 24
Therefore m = 24
The unique solution is m = 24 and n = 1.
Answer: c) The question can be answered using both statements together, but cannot be answered using either statement alone.
