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Zorluk: OrtaNumber Properties and Integer Constraints in Data Sufficiency

If xx and yy are positive integers, is x3−y3x^3 - y^3 divisible by 33?

(1) x+2yx + 2y is divisible by 33.
(2) x2−y2x^2 - y^2 is divisible by 33.

  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.
The question asks whether x3−y3x^3 - y^3 is divisible by 3. By Fermat's Little Theorem or factoring a3−a=a(a−1)(a+1)a^3 - a = a(a-1)(a+1) (the product of three consecutive integers, always divisible by 3), any integer cubed has the same remainder when divided by 3 as the integer itself (a3≡a(mod3)a^3 \equiv a \pmod 3). Therefore, x3−y3x^3 - y^3 is a multiple of 3 if and only if x−yx - y is a multiple of 3. Statement (1) specifies that x+2yx + 2y is divisible by 3. We can rewrite x+2yx + 2y as (x−y)+3y(x - y) + 3y. Because 3y3y is automatically a multiple of 3, (x−y)(x - y) must also be divisible by 3. This yields a definitive 'Yes', so Statement (1) alone is sufficient. Statement (2) states that x2−y2=(x−y)(x+y)x^2 - y^2 = (x - y)(x + y) is divisible by 3. Because 3 is prime, this requires 3 to divide either (x−y)(x - y) or (x+y)(x + y). If x=4,y=1x = 4, y = 1, 3 divides (x−y)=3(x - y) = 3, giving a 'Yes'. If x=2,y=1x = 2, y = 1, 3 divides (x+y)=3(x + y) = 3 while (x−y)=1(x - y) = 1, giving x3−y3=7x^3 - y^3 = 7, which is a 'No'. Hence, Statement (2) alone is not sufficient.

Adım Adım Çözüm

1
Rephrase the question stem using algebraic modular properties.
Since a3≡a(mod3)a^3 \equiv a \pmod 3 for any integer aa, x3−y3≡x−y(mod3)x^3 - y^3 \equiv x - y \pmod 3. Thus, x3−y3x^3 - y^3 is divisible by 3 if and only if x−yx - y is divisible by 3.
Simplifying the target expression reduces the problem to determining whether 3 divides (x−y)(x - y).
2
Evaluate Statement (1): x+2yx + 2y is divisible by 3.
x+2y=(x−y)+3yx + 2y = (x - y) + 3y. Since 3y3y is always a multiple of 3, (x+2y)(x + 2y) is divisible by 3 if and only if (x−y)(x - y) is divisible by 3.
This guarantees a definitive 'Yes' to the target question. Statement (1) alone is SUFFICIENT.
3
Evaluate Statement (2): x2−y2x^2 - y^2 is divisible by 3.
x2−y2=(x−y)(x+y)x^2 - y^2 = (x - y)(x + y). Since 3 is prime, 3 must divide (x−y)(x - y) or (x+y)(x + y). If x=4,y=1x = 4, y = 1, then x2−y2=15x^2 - y^2 = 15 (divisible by 3) and x3−y3=63x^3 - y^3 = 63 (divisible by 3) -> YES. If x=2,y=1x = 2, y = 1, then x2−y2=3x^2 - y^2 = 3 (divisible by 3), but x3−y3=7x^3 - y^3 = 7 (NOT divisible by 3) -> NO.
Statement (2) yields both 'Yes' and 'No' cases. Statement (2) alone is NOT SUFFICIENT.

Anahtar Kavram

Divisibility and Modular Congruences in Integer Properties
Tahmini Süre:2m 0s
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Konular

Number Properties and Integer Constraints in Data Sufficiency

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