Question #1 is solved here - https://www.beatthegmat.com/n-y-are-posi ... 52635.html
Question #2 is solved here - https://www.beatthegmat.com/area-of-circ ... 85340.html
Question #3 is solved here - https://www.beatthegmat.com/sum-a1-a2-a3-an-t109893.html

Please let me know if need any further help!

Anurag@Gurome wrote:

intenseCK wrote:Q1. N & Y are positive integers and 450y=n^3, which is an integer?
y/3*2^2*5
y/3^2*2*5
y/3*2*5^2
A. None
B.I
C. II
d.III
e. I, II, III

450y=n^3 implies (3^2 * 5^2 * 2)y = n^3, which implies y should be 2^2 * 3 * 5 so that 450y=n^3
Then (I) will be an integer.
(II) and (III) will not be an integer always.

The correct answer is B.

Hi Anurag- can u elaborate a little more as i am struggling to get your logic. why shd Y be 2^2*3*5

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 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:
450y = 2 * 3² * 5² * y

Since 450 provides only one 2, two 3's, and two 5's, y must provide the missing prime factors. We need y to provide two more 2's, one more 3, and one more 5.
Thus, the smallest possible value is y = 2² * 3 * 5.

Onto the answer choices:

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.
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.
Eliminate every remaining answer choice that includes II.
Eliminate E.

The correct answer is B.