UPSC CSE Prelims 2024

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?

EXPLANATION

The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone

Analysis of Statement-I: The condition m + n > mn, for natural numbers m and n, implies that at least one of the numbers must be 1. If both m ≥ 2 and n ≥ 2, then (m-1)(n-1) ≥ 1, which expands to mn - m - n + 1 ≥ 1, or mn ≥ m + n. This contradicts the given m + n > mn. Since the problem also states m > n, it follows that n must be 1. However, Statement-I alone does not provide a unique value for m (any natural number m > 1 would satisfy m + 1 > m). Therefore, Statement-I alone is insufficient to determine the values of m and n.
Analysis of Statement-II: The condition mn = 24 provides multiple pairs of natural numbers (m, n) whose product is 24. Examples include (1, 24), (2, 12), (3, 8), (4, 6), (6, 4), (8, 3), (12, 2), (24, 1). This statement alone does not uniquely determine the values of m and n. Therefore, Statement-II alone is insufficient to answer the question.
Combining Statement-I and Statement-II: From Statement-I, we established that n = 1. Substituting n = 1 into the equation from Statement-II (mn = 24), we get m * 1 = 24, which yields m = 24. This provides a unique solution (m=24, n=1) that satisfies all given conditions: m and n are natural numbers, m > n (24 > 1), m + n > mn (24 + 1 > 24*1, i.e., 25 > 24), and mn = 24. Thus, both statements together are sufficient to answer the question.

Option 1: This option is incorrect because, as demonstrated, neither Statement-I nor Statement-II alone is sufficient to uniquely determine the values of m and n.
Option 2: This option is incorrect because, as shown in the individual analyses, neither statement individually provides a unique solution for m and n.
Option 4: This option is incorrect because combining the information from both statements leads to a unique determination of m=24 and n=1, thereby answering the question.