Test the SMALLEST POSSIBLE CASE.If n and y are positive integers and 450y = n³, which of the following must be an integer?
I. y/(3 x 2² x 5)
II. y/(3² x 2 x 5)
III. y/(3 x 2 x 5²)
a. None
b. I only
c. II only
d. III only
e. I, II, and III
450y = n³ implies that 450y is the cube of an integer.
When we prime-factorize the cube of an integer, we get 3 (or a multiple of 3) of every prime factor:
8 is the cube of an integer because 8 = 2³ = 2*2*2.
27 is the cube of an integer because 27 = 3³ = 3*3*3.
Thus, when we prime-factorize 450y, we need to get AT LEAST 3 of every prime factor.
Here's the prime-factorization of 450y:
450y = 2 * 3² * 5² * y
Since 450 provides only one 2, two 3's, and two 5's, and we need at least 3 of every prime factor, the missing prime factors must be provided by y.
Thus, y must provide at at least two more 2's, one more 3, and one more 5.
SMALLEST POSSIBLE CASE:
y = 2² * 3 * 5.
Test y = 2² * 3 * 5 in statements I, II and III.
I. y/(3 x 2² x 5)
(2² * 3 * 5)/(3 x 2² x 5) = 1.
The smallest possible value of y yields an integer.
Thus, statement I must be true.
Eliminate every answer choice that does not include I.
Eliminate A, C and D.
II. y/(3² x 2 x 5)
(2² * 3 * 5)/(3² x 2² x 5) = 1/3.
Not an integer.
Thus, statement II does NOT have to be true.
Eliminate every remaining answer choice that includes II.
Eliminate E.
The correct answer is B.
