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If n and y are positive integers and 450y = n³, which of the following must be an integer?"
I. [tex]\frac{y}{3*2^2*5}[/tex]
II. [tex]\frac{y}{3^2*2*5}[/tex]
III. [tex]\frac{y}{3*2*5^2}[/tex]
(A) None
(B) I only
(C) II only
(D) III only
(E) I, II, and III

Respuesta :

Answer:

(B) I only

Step-by-step explanation:

450y = n³

y = n³ / 450 = n³ / (3² * 2 * 5²)

in order to keep y and n be positive integer, the minimal requirement for n³ is n³ = (3³ * 2³ * 5³)

y = n³ / 450

  = n³ / (3² * 2 * 5²)

  = (3³ * 2³ * 5³) / (3² * 2 * 5²)

  = 3*2²*5

∴ I. y / (3*2²*5) =  ((3³ * 5³ * 2³) / (3² * 5² * 2)) / (3*2²*5) = 1 ... that keep answer as the smallest positive integer .... Correct answer

II.   y / (3²*2*5) =  ((3³ * 5³ * 2³) / (3² * 5² * 2)) / (3²*2*5) = 2/3 ...not integer

III.  y / (3²*2*5) =  ((3³ * 5³ * 2³) / (3² * 5² * 2)) / (3*2*5²) = 2/5 ...not integer

Answer: The correct answer is neither

Step-by-step explanation:

for DeltaMath.

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