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 provides the inequalities m + n > mn and m > n. For natural numbers, if m ≥ 2 and n ≥ 2, then mn ≥ m + n, which contradicts the first inequality. Therefore, one of the numbers must be 1. Since m > n, n must be 1, and m can be any natural number greater than 1. This does not give a unique value for m, so Statement I alone is insufficient.

• Statement II states that mn = 24. The pairs of natural numbers (m, n) that satisfy this are (1, 24), (2, 12), (3, 8), (4, 6), (6, 4), (8, 3), (12, 2), and (24, 1). There are multiple possible values for m and n, so Statement II alone is insufficient.

• Combining both statements, we use the result from Statement I that n must be 1. Substituting n = 1 into Statement II (mn = 24) gives m × 1 = 24, so m = 24.

• We verify that m = 24 and n = 1 satisfy both conditions: m + n = 25 > 24 = mn, and m = 24 > 1 = n. The values are unique.

• Since neither statement alone is sufficient but both together yield a unique solution, the correct option is (c).

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 is insufficient because it only restricts n to 1 but allows any m > 1, and Statement II alone is insufficient because it allows multiple factor pairs of 24.
(b) The Question can be answered by using either Statement alone: Neither statement alone is sufficient; Statement I leaves m undetermined, and Statement II has multiple solutions for m and n.
(d) The Question cannot be answered even by using both the Statements together: The question can be answered uniquely by combining both statements, which gives m = 24 and n = 1, so it is not impossible to answer.

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.