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Difficulty: MediumStatement Independence Evaluation and Statement Combination

If mm and nn are positive integers, what is the value of m+nm + n?

(1) m2−n2=21m^2 - n^2 = 21
(2) m−n=3m - n = 3

  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.
The correct option identifies that neither statement alone provides enough information to determine a single value for m+nm + n, but combining both statements allows us to substitute m−n=3m - n = 3 directly into the factored form (m−n)(m+n)=21(m - n)(m + n) = 21, yielding m+n=7m + n = 7 uniquely.

Step-by-Step Solution

1
Analyze the question target and algebraic constraints
The target is to find a unique value for m+nm + n, where mm and nn are positive integers.
Establishing positive integer constraints is critical when evaluating factor possibilities.
2
Evaluate Statement (1) independently
Statement (1) states m2−n2=21m^2 - n^2 = 21, which factors to (m−n)(m+n)=21(m - n)(m + n) = 21. Positive integer factor pairs for 2121 are (1,21)(1, 21) and (3,7)(3, 7). If m−n=1m - n = 1 and m+n=21m + n = 21, then m+n=21m + n = 21. If m−n=3m - n = 3 and m+n=7m + n = 7, then m+n=7m + n = 7. Two values are possible, so Statement (1) alone is INSUFFICIENT.
A single unique value cannot be determined from Statement (1) alone.
3
Evaluate Statement (2) independently without carrying over Statement (1) data
Statement (2) states m−n=3m - n = 3. Infinitely many positive integer pairs (m,n)(m, n) satisfy this equation (e.g., (4,1)  ⟹  m+n=5(4, 1) \implies m+n=5; (5,2)  ⟹  m+n=7(5, 2) \implies m+n=7). Thus, Statement (2) alone is INSUFFICIENT.
Statement (2) must be evaluated strictly isolated from Statement (1).
4
Combine Statement (1) and Statement (2)
Substitute m−n=3m - n = 3 into (m−n)(m+n)=21(m - n)(m + n) = 21: 3(m+n)=21  ⟹  m+n=73(m + n) = 21 \implies m + n = 7. A single unique value is determined.
Combining both statements resolves the ambiguity present in each statement individually.

Key Concept

Statement Independence Evaluation and Algebraic Combination in Data Sufficiency
Estimated Time:1m 30s
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Topics

Statement Independence Evaluation and Statement Combination

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