Question:
If \(m,n\) are distinct integers, what is the value of \((m-n)^2\)?
Statements:
(I) \(m,n\) are prime numbers.
(II) \(30 \leq m,n \leq 40\)
If \(m,n\) are distinct integers, what is the value of \((m-n)^2\)?
Statements:
(I) \(m,n\) are prime numbers.
(II) \(30 \leq m,n \leq 40\)
Show Hint
In data sufficiency, if the order changes but the squared value remains same, it is still uniquely determined.
Updated On: Jul 15, 2026
- Statement (I) alone is sufficient to answer the question, but (II) alone is not sufficient.
- Statement (II) alone is sufficient to answer the question, but (I) alone is not sufficient.
- Both the statements (I) and (II) are sufficient to answer the question, but neither statement alone is sufficient.
- Both the statements (I) and (II) together are not sufficient to answer the question and additional data are required.
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The Correct Option is C
Solution and Explanation
Concept:
To determine the exact value of:
\[
(m-n)^2
\]
we need the exact difference between \(m\) and \(n\).
Step 1: Checking Statement (I).
Given: \[ m,n \text{ are prime numbers} \] Possible cases: \[ m=3,\; n=5 \] \[ (m-n)^2=(3-5)^2=4 \] Another case: \[ m=7,\; n=13 \] \[ (m-n)^2=(7-13)^2=36 \] Different values are possible. So, Statement (I) alone is not sufficient.
Step 2: Checking Statement (II).
Given: \[ 30 \leq m,n \leq 40 \] Possible distinct integer pairs: \[ m=31,\; n=32 \] \[ (m-n)^2=1 \] Another case: \[ m=35,\; n=40 \] \[ (m-n)^2=25 \] Different values are possible. So, Statement (II) alone is not sufficient.
Step 3: Checking both statements together.
Prime numbers between \(30\) and \(40\) are: \[ 31,\; 37 \] Since \(m,n\) are distinct prime numbers in this range, the only possible values are: \[ m=31,\; n=37 \] or \[ m=37,\; n=31 \] Thus: \[ m-n=\pm 6 \] So: \[ (m-n)^2=36 \] This gives a unique value. Hence, both statements together are sufficient.
Step 1: Checking Statement (I).
Given: \[ m,n \text{ are prime numbers} \] Possible cases: \[ m=3,\; n=5 \] \[ (m-n)^2=(3-5)^2=4 \] Another case: \[ m=7,\; n=13 \] \[ (m-n)^2=(7-13)^2=36 \] Different values are possible. So, Statement (I) alone is not sufficient.
Step 2: Checking Statement (II).
Given: \[ 30 \leq m,n \leq 40 \] Possible distinct integer pairs: \[ m=31,\; n=32 \] \[ (m-n)^2=1 \] Another case: \[ m=35,\; n=40 \] \[ (m-n)^2=25 \] Different values are possible. So, Statement (II) alone is not sufficient.
Step 3: Checking both statements together.
Prime numbers between \(30\) and \(40\) are: \[ 31,\; 37 \] Since \(m,n\) are distinct prime numbers in this range, the only possible values are: \[ m=31,\; n=37 \] or \[ m=37,\; n=31 \] Thus: \[ m-n=\pm 6 \] So: \[ (m-n)^2=36 \] This gives a unique value. Hence, both statements together are sufficient.
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