Hi,
450=2.3^2.5^2
So, 2.3^2.5^2.y = n^3
So, y is of the form 2^2.3.5.p^3 where p is an integer
So, y/3*2^2*5 is an integer.

[quote="Frankenstein"]Hi,
450=2.3^2.5^2
So, 2.3^2.5^2.y = n^3
So, y is of the form 2^2.3.5.p^3 where p is an integer
So, y/3*2^2*5 is an integer.[/quote]

Good one!!

LHS tells us its a cube, equate both sides. As RHS is not a cube make it and find Y which comes out to be 2^2*3*5, so it must be divisible by it.

Advice- this is a symmetry based problem look for symmetry on two sides of so formed equations.

Frankenstein wrote:Hi,
450=2.3^2.5^2
So, 2.3^2.5^2.y = n^3
So, y is of the form 2^2.3.5.p^3 where p is an integer
So, y/3*2^2*5 is an integer.

Hey i agree with IMO A
But option D is also correct
y/3*2*5

cans wrote:

Frankenstein wrote:Hi,
450=2.3^2.5^2
So, 2.3^2.5^2.y = n^3
So, y is of the form 2^2.3.5.p^3 where p is an integer
So, y/3*2^2*5 is an integer.

Hey i agree with IMO A
But option D is also correct
y/3*2*5

Hi,
Yes.. D is correct too. But, I haven't really checked other options after getting A. Moreover, those * and ^ were so awkward, I wanted to skip this question asap!

hey guyzzz... i did not at all understand the solution of this sum > N and Y are +ve integers and 450y = n^3 which of the following is an integer?

y/3*2^2*5
y/3^2*2*5
y/3*2*5^2
y/3*2*5
y/3^3*2^2*5

Can anybody please help me to really understand this sum.....

divya23 wrote:N and Y are +ve integers and 450y = n^3 which of the following is an integer?

y/3*2^2*5
y/3^2*2*5
y/3*2*5^2
y/3*2*5
y/3^3*2^2*5

[spoiler]oa = y/3*2^2*5[/spoiler]

3*3*5*5*2*y = n^3
to make n a perfect cube, y should be 3*5*2^2
hence option A

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.
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 3 of every prime factor, the missing prime factors must be provided by y.
Thus, y must provide two more 2's, one more 3, and one more 5.
Thus, the smallest possible value of 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.

450 = 2* 3^2 * 5^2

thus 3*2^2*5 is required = 60

y/60 = integer also y/30 too.

A and D