Examkin

Soru

Zorluk: OrtaQuestion Stem Simplification and Target Rephrasing

If mm and nn are positive integers such that m≠nm \neq n, is m2+mn−2n2m−n\frac{m^2 + mn - 2n^2}{m - n} an even integer?

(1) m+3m + 3 is an odd integer.
(2) nn is an odd integer.

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Cevap
  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. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
By factoring the numerator of the expression in the question stem, m2+mn−2n2m−n=(m+2n)(m−n)m−n=m+2n\frac{m^2 + mn - 2n^2}{m - n} = \frac{(m + 2n)(m - n)}{m - n} = m + 2n. Since 2n2n is always even, m+2nm + 2n is even if and only if mm is even. Rephrasing the question target yields: "Is mm an even integer?" Statement (1) tells us m+3m + 3 is odd, which means mm must be even (giving a definitive "Yes" answer). Statement (2) provides information about nn, which is irrelevant to whether mm is even. Thus, Statement (1) alone is sufficient, but Statement (2) alone is not.

Adım Adım Çözüm

1
Simplify the target expression in the question stem
m2+mn−2n2m−n=(m+2n)(m−n)m−n=m+2n\frac{m^2 + mn - 2n^2}{m - n} = \frac{(m + 2n)(m - n)}{m - n} = m + 2n
Factoring the numerator simplifies the target expression for m≠nm \neq n.
2
Rephrase the target question using parity properties
Since 2n2n is an even integer for any integer nn, m+2nm + 2n is even if and only if mm is an even integer. The rephrased question target is: "Is mm an even integer?"
Simplifying the target question eliminates the variable nn from the requirement.
3
Evaluate Statement (1): m+3m + 3 is an odd integer
If m+3m + 3 is odd, then m=odd−3=evenm = \text{odd} - 3 = \text{even}. Thus, mm is definitively even. Statement (1) alone is SUFFICIENT.
Subtracting an odd integer from an odd integer yields an even integer.
4
Evaluate Statement (2): nn is an odd integer
Statement (2) gives information about nn, but gives no information about mm. Since the rephrased question depends solely on mm, Statement (2) alone is INSUFFICIENT.
Knowing nn does not determine whether mm is even.

Anahtar Kavram

Algebraic factoring and target rephrasing in Data Sufficiency questions
Bu soruyu puanla

Konular

Question Stem Simplification and Target Rephrasing

Bu konudan daha fazla soru