CSAT Solved Papers/ 2024/Q59

2024 CSAT — Q59

Quant Data sufficiency 2.5 marks Hard

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 Answer
D The Question cannot be answered even by using both the Statements together

Worked rationale

Decide, don’t just compute, and test each statement for uniqueness.

Statement-I alone ( $m+n > mn$ with $m>n$ ): rearrange $m+n>mn \iff (m-1)(n-1) < 1$ . For naturals $(m-1)(n-1)$ is a non-negative integer, so it must be $0$ , forcing $m=1$ or $n=1$ . With $m>n\ge1$ , this gives $n = 1$ and $m$ any integer $> 1$ (e.g. $(2,1),(7,1),(24,1)$ all satisfy I). Not unique.

Statement-II alone ( $mn = 24$ ): factor pairs $(1,24),(2,12),(3,8),(4,6),\dots$ — many. Not unique.

Both together. From I, $n = 1$ ; from II, $m\cdot1 = 24 \Rightarrow m = 24$ . Check: $m>n$ ( $24>1$ ✓) and $m+n = 25 > mn = 24$ ✓. Unique solution $(m,n) = (24,1)$ .

Answer: (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.

Why the other options miss

A

thought a statement was enough when it wasn’t: thinks Statement I pins the pair, missing that $n=1$ leaves $m$ free.
B

thought either statement was enough when neither is: treats $mn=24$ as if it had a unique factorisation, or reads I as fully determining.
D

failed to extract the key deduction: fails to extract $n=1$ from I, so it never sees that II then forces $m=24$ uniquely.

Specialist insight

The decisive algebra is $m+n>mn \iff (m-1)(n-1)<1$ , which over the naturals collapses to “one of them is $1$ .” That single deduction is what makes the statements complementary: I alone fixes $n=1$ but not $m$ ; II alone gives the product but not which factor pair; together, $n=1$ selects the pair $(24,1)$ uniquely. The trap is treating either statement as self-sufficient — always test a statement by trying to produce a second admissible solution; here I clearly admits many, II admits many, but their intersection is a single point.

The trap, in one line

$m+n>mn \iff (m-1)(n-1)<1$ forces $n=1$ (from I), then $mn=24$ (from II) pins $m=24$ — neither alone is unique, both together are: $(c)$ .

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