If n and y are positive integers and 450y = n³, which of the following must be an integer? "
i.
ii.
iii.
(a) none
(b) i only
(c) ii only
(d) iii only
(e) i, ii, and iii

(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

0.02040816326