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Difficulty: Very hardNumber Properties and Integer Constraints in Data Sufficiency

If mm and nn are positive real numbers, is m+nm + n an integer?

(1) m2−n2m^2 - n^2 is a prime number.
(2) m−nm - n is a positive integer.

  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. C
    BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient.
  4. D
    EACH statement ALONE is sufficient.
  5. Statements (1) and (2) TOGETHER are NOT sufficient.Answer

Answer

Statements (1) and (2) together are NOT sufficient.
The correct answer is that both statements combined are insufficient. A common high-level trap in GMAT Data Sufficiency is assuming variables must be integers when the stem specifies 'real numbers'. Combining (m−n)(m+n)=p(m - n)(m + n) = p (prime) with m−n=km - n = k (positive integer) gives m+n=pkm + n = \frac{p}{k}. If m=2.25m = 2.25 and n=0.25n = 0.25, we get m−n=2m - n = 2 (integer) and m2−n2=5m^2 - n^2 = 5 (prime), but m+n=2.5m + n = 2.5 (not an integer). Conversely, m=2m = 2 and n=1n = 1 gives m−n=1m - n = 1 and m2−n2=3m^2 - n^2 = 3 (prime), where m+n=3m + n = 3 (an integer). Because both an affirmative and a negative answer are possible, the information remains insufficient.

Step-by-Step Solution

1
Evaluate Statement (1) independently.
Statement (1) is NOT sufficient.
If m2−n2=2m^2 - n^2 = 2 (a prime number), we could have m=2m = 2 and n=√2n = √2, giving m+n=2+√2m + n = 2 + √2 (not an integer). Alternatively, m=2m = 2 and n=1n = 1 gives m2−n2=3m^2 - n^2 = 3 (prime) and m+n=3m + n = 3 (an integer). Hence, Statement (1) alone is insufficient.
2
Evaluate Statement (2) independently.
Statement (2) is NOT sufficient.
If m−n=2m - n = 2, we could have m=3.5m = 3.5 and n=1.5n = 1.5, giving m+n=5m + n = 5 (an integer). Or we could have m=3.2m = 3.2 and n=1.2n = 1.2, giving m+n=4.4m + n = 4.4 (not an integer). Hence, Statement (2) alone is insufficient.
3
Evaluate Statements (1) and (2) combined.
Test for consistency with both integer and non-integer outcomes for m+nm + n.
We know m2−n2=(m−n)(m+n)=pm^2 - n^2 = (m - n)(m + n) = p, where pp is prime and m−n=km - n = k is a positive integer. Thus k(m+n)=pk(m + n) = p, so m+n=pkm + n = \frac{p}{k}.
4
Construct concrete counterexamples under both combined statements.
Statements (1) and (2) together are NOT sufficient.
Case 1 (YES): Let m=2m = 2 and n=1n = 1. Then m2−n2=3m^2 - n^2 = 3 (prime) and m−n=1m - n = 1 (positive integer). Here, m+n=3m + n = 3, which IS an integer.
Case 2 (NO): Let m=2.25m = 2.25 and n=0.25n = 0.25. Both are positive real numbers. m−n=2.25−0.25=2m - n = 2.25 - 0.25 = 2 (positive integer). m2−n2=(2.25−0.25)(2.25+0.25)=2×2.5=5m^2 - n^2 = (2.25 - 0.25)(2.25 + 0.25) = 2 \times 2.5 = 5 (prime number). But m+n=2.25+0.25=2.5m + n = 2.25 + 0.25 = 2.5, which is NOT an integer. Since m+nm + n can be an integer or a non-integer, both statements together are not sufficient.

Key Concept

Implicit Integer Constraints vs. Real Number Definitions in Data Sufficiency
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Topics

Number Properties and Integer Constraints in Data Sufficiency

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