integers

[daretodream]

If n and y are positive integers, and 450y=n^3 , which of the following is 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

[thephoenix]

If n and y are positive integers, and 450y=n^3 , which of the following is 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

450 y = n^3

450 = 3*3*5*5*2

in order for 450 to have a cube root we need 3 of each number of 3*3*3*5*5*5*2*2*2

we already have two 3's and two 5's and one 2

therefore we need one 3, one 5, and 2 twos

or 3 x 5 x 2^2

that would give answer I

[harsh.champ]

If n and y are positive integers, and 450y=n^3 , which of the following is 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

450 y = n^3

450 = 3*3*5*5*2

in order for 450 to have a cube root we need 3 of each number of 3*3*3*5*5*5*2*2*2

we already have two 3's and two 5's and one 2

therefore we need one 3, one 5, and 2 twos

or 3 x 5 x 2^2

that would give answer I

Hey thephoenix,
I think you could have done a mistake over here.
The correct answer is B but in haste you could have chosen the answer as A as in your answer you had written:-

that would give answer I

It is a similar mistake that I did in the probability question and didn't consider the sample space.

[ajith]

If n and y are positive integers, and 450y=n^3 , which of the following is 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

450y=n^3
2*3^2*5^2*y = n^3

n^3 has all the prime factors with powers multiples of three

so y should have 2^2*5*3 as factor (at least)

so y / ( 3 * 2^2 * 5 ) is an integer, others may or may not be an integer

B

[shashank.ism]

If n and y are positive integers, and 450y=n^3 , which of the following is 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

450 = 3x3x5x5x2
450 y = n^3 --> n = cube root(450 y)
so for 450 y to be a perfect cube , we need to add more 3, 5 and 2 as a factor 0f y

[spoiler]so y is a multiple of 3x5 x 2^2

so only I i.e. B[/spoiler]

[thephoenix]

If n and y are positive integers, and 450y=n^3 , which of the following is 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

450 y = n^3

450 = 3*3*5*5*2

in order for 450 to have a cube root we need 3 of each number of 3*3*3*5*5*5*2*2*2

we already have two 3's and two 5's and one 2

therefore we need one 3, one 5, and 2 twos

or 3 x 5 x 2^2

that would give answer I

WHICH IS OPTION B