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Zorluk: ZorNumber Properties and Integer Constraints in Data Sufficiency

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

(1) m−nm - n is an integer.
(2) m2−n2m^2 - n^2 is an 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.Cevap

Cevap

Statements (1) and (2) together are not sufficient.
Evaluating Statement (1) alone: If m=2m = \sqrt{2} and n=2n = \sqrt{2}, then m−n=0m - n = 0 (an integer), but m+n=22m + n = 2\sqrt{2} is not an integer. If m=2m = 2 and n=1n = 1, then m−n=1m - n = 1 (an integer) and m+n=3m + n = 3 is an integer. Thus, Statement (1) alone is not sufficient.

Evaluating Statement (2) alone: If m=2m = \sqrt{2} and n=1n = 1, then m2−n2=1m^2 - n^2 = 1 (an integer), but m+n=2+1m + n = \sqrt{2} + 1 is not an integer. If m=2m = 2 and n=1n = 1, then m2−n2=3m^2 - n^2 = 3 (an integer) and m+n=3m + n = 3 is an integer. Thus, Statement (2) alone is not sufficient.

Evaluating Statements (1) and (2) together: Factoring gives m2−n2=(m−n)(m+n)m^2 - n^2 = (m - n)(m + n). If m−n=k≠0m - n = k \neq 0 and m2−n2=pm^2 - n^2 = p for integers kk and pp, then m+n=pkm + n = \frac{p}{k}. This establishes that m+nm + n must be rational, but a rational number is not necessarily an integer. For example, if m=54m = \frac{5}{4} and n=−34n = -\frac{3}{4}, then m−n=2m - n = 2 (an integer) and m2−n2=1m^2 - n^2 = 1 (an integer), but m+n=12m + n = \frac{1}{2} is not an integer. If m=2m = 2 and n=0n = 0, both statements hold and m+n=2m + n = 2 is an integer. Since m+nm + n can be an integer or a non-integer, both statements together are not sufficient.

Adım Adım Çözüm

1
Analyze the question stem and identify variable constraints.
The variables mm and nn are defined as real numbers, NOT necessarily integers.
Failing to account for non-integer real values is a primary trap in GMAT Data Sufficiency number property questions.
2
Evaluate Statement (1) independently.
Statement (1) states m−n=km - n = k for some integer kk.
If m=2m = 2 and n=1n = 1, m−n=1m - n = 1 (integer) and m+n=3m + n = 3 (integer).
If m=2m = \sqrt{2} and n=2n = \sqrt{2}, m−n=0m - n = 0 (integer) and m+n=22m + n = 2\sqrt{2} (not an integer).
Since m+nm + n can be an integer or non-integer, Statement (1) is NOT sufficient.
Testing both integer and irrational values reveals that a difference being an integer does not constrain the sum to be an integer.
3
Evaluate Statement (2) independently.
Statement (2) states m2−n2=pm^2 - n^2 = p for some integer pp.
If m=2m = 2 and n=1n = 1, m2−n2=3m^2 - n^2 = 3 (integer) and m+n=3m + n = 3 (integer).
If m=2m = \sqrt{2} and n=1n = 1, m2−n2=1m^2 - n^2 = 1 (integer) and m+n=2+1m + n = \sqrt{2} + 1 (not an integer).
Since m+nm + n can be an integer or non-integer, Statement (2) is NOT sufficient.
Testing non-integer values shows that the difference of squares being an integer does not guarantee that the sum is an integer.
4
Evaluate Statements (1) and (2) together.
We have m−n=km - n = k and (m−n)(m+n)=p(m - n)(m + n) = p for integers kk and pp.
If k=0k = 0, m=nm = n, so m−n=0m - n = 0 and m2−n2=0m^2 - n^2 = 0. In this case m+n=2mm + n = 2m, which can be non-integer if m=2m = \sqrt{2}.
If k≠0k \neq 0, then m+n=pkm + n = \frac{p}{k}. This proves that m+nm + n is a rational number, but not necessarily an integer.
For instance, let m=54m = \frac{5}{4} and n=−34n = -\frac{3}{4}:
- m−n=54−(−34)=2m - n = \frac{5}{4} - \left(-\frac{3}{4}\right) = 2 (integer)
- m2−n2=2516−916=1m^2 - n^2 = \frac{25}{16} - \frac{9}{16} = 1 (integer)
- m+n=54+(−34)=12m + n = \frac{5}{4} + \left(-\frac{3}{4}\right) = \frac{1}{2} (not an integer).
Conversely, if m=2m = 2 and n=0n = 0, m−n=2m - n = 2 (integer), m2−n2=4m^2 - n^2 = 4 (integer), and m+n=2m + n = 2 (integer).
Because m+nm + n can still be an integer or a non-integer, both statements together are NOT sufficient.
Algebraic division demonstrates that the sum is guaranteed to be rational, but rational numbers include non-integer fractions.

Anahtar Kavram

Real Number Constraints vs. Integer Constraints in Data Sufficiency
Tahmini Süre:2m 0s
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Number Properties and Integer Constraints in Data Sufficiency

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