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Difficulty: MediumNumber Properties and Integer Constraints in Data Sufficiency

If mm and nn are positive integers, is m2−n2m^2 - n^2 a prime number?

(1) m−n=1m - n = 1
(2) m+n=7m + n = 7

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Answer
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Rephrasing the question stem shows m2−n2=(m−n)(m+n)m^2 - n^2 = (m - n)(m + n). Statement (1) alone allows 2n+12n + 1 to be prime (for n=1n=1) or composite (for n=4n=4), so it is insufficient. Statement (2) alone allows pairs like (4,3)(4,3) which gives 77 (prime) and (5,2)(5,2) which gives 2121 (composite), so it is insufficient. Combining both statements yields m=4m = 4 and n=3n = 3, giving m2−n2=7m^2 - n^2 = 7, which definitively answers 'Yes'.

Step-by-Step Solution

1
Rephrase the question stem using algebraic factorization.
m2−n2=(m−n)(m+n)m^2 - n^2 = (m - n)(m + n). For m2−n2m^2 - n^2 to be a prime number, since mm and nn are positive integers, the smaller factor (m−n)(m - n) must equal 1 and the larger factor (m+n)(m + n) must be a prime number.
Factoring highlights the conditions required for a product of integers to yield a prime number.
2
Evaluate Statement (1): m−n=1m - n = 1.
Substitute m=n+1m = n + 1 into m2−n2m^2 - n^2: (1)(2n+1)=2n+1(1)(2n + 1) = 2n + 1. If n=1n = 1, then 2(1)+1=32(1) + 1 = 3 (prime, Yes). If n=4n = 4, then 2(4)+1=92(4) + 1 = 9 (composite, No).
Since Statement (1) produces both Yes and No outcomes, it is NOT sufficient.
3
Evaluate Statement (2): m+n=7m + n = 7.
Since mm and nn are positive integers, test pairs with m>nm > n: If (m,n)=(4,3)(m, n) = (4, 3), then m2−n2=16−9=7m^2 - n^2 = 16 - 9 = 7 (prime, Yes). If (m,n)=(5,2)(m, n) = (5, 2), then m2−n2=25−4=21m^2 - n^2 = 25 - 4 = 21 (composite, No).
Since Statement (2) produces both Yes and No outcomes, it is NOT sufficient.
4
Evaluate Statement (1) and Statement (2) together.
System of equations: m−n=1m - n = 1 and m+n=7m + n = 7. Adding the equations gives 2m=8  ⟹  m=42m = 8 \implies m = 4, which leads to n=3n = 3. Then m2−n2=42−32=7m^2 - n^2 = 4^2 - 3^2 = 7, which is prime (Definitive Yes).
Combining the statements yields a single unique pair of values and a definitive Yes answer.

Key Concept

Difference of squares factorization and prime number properties in Data Sufficiency
Estimated Time:2m 0s
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Topics

Number Properties and Integer Constraints in Data Sufficiency

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