Question

If n and y are positive integers and 450y = n³, which of the following must be an integer?"

I.
`(y)/(3*2^2*5)`
II.
`(y)/(3^2*2*5)`
III.
`(y)/(3*2*5^2)`
(A) None
(B) I only
(C) II only
(D) III only
(E) I, II, and III

Answer 1

Answer: The correct answer is neither

Explanation:

for DeltaMath.

Answer 2

(B) I only

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