moneyman wrote:I dont know what is the component I should be focussing on while doing this problem

If n and y are positive integers and 450y=n^3, 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

Thanks

450 = 2 x 5^2 x 3^2
Now 2 x 5^2 x 3^2 x y = n^3
So at minimum, y = 2^2 x 5 x 3
So y/3*2^2*5 is the minimum value which will always be integer. Got me, Max?

Got it Amitava..But I really could not figure out the approach..now its clear..Thanks

i tried to push numbers out for this and wound up with n=10 and y=2.2

when i plugged them into the answer choices not one of the roman numerals gave an integer. So my first answer is none will give an integer.


I tried to break this up into primes but then got lost.

can someone explain this to me in baby steps and please give qa. thanks folks!

Enginpasa1 wrote:i tried to push numbers out for this and wound up with n=10 and y=2.2

when i plugged them into the answer choices not one of the roman numerals gave an integer. So my first answer is none will give an integer.


I tried to break this up into primes but then got lost.

can someone explain this to me in baby steps and please give qa. thanks folks!

Step by step:

Find the prime factors of 450.
450
->45*10
->9*5*2*5
-->3*3*5*2*5
--->2*3*3*5*5

The question states that 450y = n^3.
Since y and n are both positive integers, this means that 450y must be a perfect cube.

In order for 450y to be a perfect cube, we look at our current prime factors of 450:
(2) (3*3) (5*5)

Therefore, in order for 450y to be a perfect cube, we need an additional 5, an additional 3, and two additional 2's.

(2) (3*3) (5*5) * y
->(2) (3*3) (5*5) * [(2*2) (3) (5)]
-->(2*2*2) (3*3*3) (5*5*5) = perfect cube

Thus, at minimum, y must be [(2*2) (3) (5)] or 2^2 * 3 * 5.

We look at our answer choices and plug in for y.

I)
y/3*2^2*5
-> (2^2 * 3 * 5) / 3*2^2*5 = 1 (integer)

II)
y/3^2*2*5
-> (2^2 * 3 * 5) / 3^2*2*5 = 2/3 (not an integer)

III)
y/3*2*5^2
-> (2^2 * 3 * 5) / 3*2*5^2 = 2/5 (not an integer)

Is there a way to approach this problem and not go through all of your steps. Can we force a value for y and n and then test each roman numeral?

Enginpasa1 wrote:Is there a way to approach this problem and not go through all of your steps. Can we force a value for y and n and then test each roman numeral?

To me, this approach makes the most sense, and like camitava showed, it is pretty straightforward. I would not go through my step by step method on paper; rather, a lot of it would be mental.

The underlying process is quite simple:
1. Recognize that 450y must be a perfect cube.
2. Break 450 into its prime factors and see what y at minimum must be to create a perfect cube.
3. Glance at I, II, and III to see if this minimum value of y can cancel all the denominator values.

i guess the answer hinges on seeing that 450y must be a perfect cube. NOw I get it! thanks

Great solution.

Enginpasa1 wrote:i tried to push numbers out for this and wound up with n=10 and y=2.2

when i plugged them into the answer choices not one of the roman numerals gave an integer. So my first answer is none will give an integer.


I tried to break this up into primes but then got lost.

can someone explain this to me in baby steps and please give qa. thanks folks!

A quick suggestion Enginpasa:

On the GMAT, more often than not, there is as much information in the options and the other parts of the question as the actual question itself.

The 3 Roman Numeral statements are CLEAR indicators that Prime factorisation is in some way or the other involved.

Any method that takes us away from that will be longer and relatively more cumbersome.

Thank you very much

tmmyc wrote:

Enginpasa1 wrote:i tried to push numbers out for this and wound up with n=10 and y=2.2

when i plugged them into the answer choices not one of the roman numerals gave an integer. So my first answer is none will give an integer.


I tried to break this up into primes but then got lost.

can someone explain this to me in baby steps and please give qa. thanks folks!

Step by step:

Find the prime factors of 450.
450
->45*10
->9*5*2*5
-->3*3*5*2*5
--->2*3*3*5*5

The question states that 450y = n^3.
Since y and n are both positive integers, this means that 450y must be a perfect cube.

In order for 450y to be a perfect cube, we look at our current prime factors of 450:
(2) (3*3) (5*5)

Therefore, in order for 450y to be a perfect cube, we need an additional 5, an additional 3, and two additional 2's.

(2) (3*3) (5*5) * y
->(2) (3*3) (5*5) * [(2*2) (3) (5)]
-->(2*2*2) (3*3*3) (5*5*5) = perfect cube

Thus, at minimum, y must be [(2*2) (3) (5)] or 2^2 * 3 * 5.

We look at our answer choices and plug in for y.

I)
y/3*2^2*5
-> (2^2 * 3 * 5) / 3*2^2*5 = 1 (integer)

II)
y/3^2*2*5
-> (2^2 * 3 * 5) / 3^2*2*5 = 2/3 (not an integer)

III)
y/3*2*5^2
-> (2^2 * 3 * 5) / 3*2*5^2 = 2/5 (not an integer)

thank you very much, I was surfing the web looking for such a good explanation. Good luck
mat