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

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If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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22 Oct 2008, 10:12
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If n and y are positive integers and 450y = n^3, which of the following must be an integer

I. $$\frac{y}{3 * 2^2 * 5}$$

II. $$\frac{y}{3^2 * 2 * 5}$$

III. $$\frac{y}{3 * 2 * 5^2}$$

A. None.
B. I only.
C. II only.
D. III only.
E. I, II, and III
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Posts: 52336
Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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24 Jan 2010, 01:34
3
5
gautamsubrahmanyam wrote:
I am not sure I fully understand the explanation.

Method1 asks for a technique to find a value of n divisible by 450.This seems to be a time consuming method
The other method I dont seem to understand.450=3^2*2*5^2
I also understand the statement "for a cube, the powers of the prime factors have to be multiples of 3"

What I dont understand is how be 3 * 2² * 5 makes each of the prime factors have powers with multiples of 3

Could someone help me on this please.

We have $$450y=2*3^2*5^2*y=n^3$$, as $$y$$ and $$n$$ are positive integers, $$y$$ must complete the powers of 2, 3 and 5 so that these powers will be the multiples of 3. Hence the least value of $$y$$ is $$2^2*3*5$$. In this case $$2*3^2*5^2*y=(2*3^2*5^2)*(2^2*3*5)=2^3*3^3*5^3=(2*3*5)^3=n^3$$.

As the least value of $$y$$ must be $$2^2*3*5$$, then $$\frac{y}{2^2*3*5}$$, will equal to 1. Choice B.

Well generally speaking (and little complicating), $$y$$ must be of a form $$y=2^{2+3p}*3^{1+3q}*5^{1+3r}*x^3$$, where $$p$$, $$q$$ and $$r$$ are the integers $$\geq0$$ and $$x$$ ($$x\geq{0}$$), is some other multiple of $$y$$ which also has the power of $$3$$.

For example $$y$$ can be $$y=2^5*3^7*5^{16}*91^3$$, in this case $$(2*3^2*5^2)*(2^5*3^7*5^{16}*91^3)=(2^2*3^3*5^6*91)^3=n^3$$.
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Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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22 Oct 2008, 10:39
19
2
rishi2377 wrote:
If n and y are positive integers and 450y = n^3, which of the following must be an interger

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

OA to follow

450y = n^3 means 450y is a cube.

450 = 3² * 2 * 5²

for a cube, the powers of the prime factors have to be multiples of 3

so if 450y is to be a cube then y has to be 3 * 2² * 5 ( to make the three prime factors have powers with multiples of 3)

so y/ 3 * 2² * 5 has to be an integer
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Joined: 05 Jul 2008
Posts: 1267
Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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22 Oct 2008, 10:36
3
rishi2377 wrote:
If n and y are positive integers and 450y = n^3, which of the following must be an interger

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

OA to follow

450y = n^3 -> y = n3/450 -> n3 is divisible by 450 & n is +ve integer. look for values of n^3 such that n is integer and n is divisible by 450. Clearly we have to look in the order of 1000's to find a value such that n3 is divisible by 450 and n is integer.

n3=1000 n =10 not divisible by 450
n3=8000 n =20 not divisible by 450
n3=27000 n =30 divisible by 450 y = 2700/45 = 60

Look for I, II , III which have denominators 60,90,150.

only I does and hence B
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Posts: 17
Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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23 Jan 2010, 23:25
I am not sure I fully understand the explanation.

Method1 asks for a technique to find a value of n divisible by 450.This seems to be a time consuming method
The other method I dont seem to understand.450=3^2*2*5^2
I also understand the statement "for a cube, the powers of the prime factors have to be multiples of 3"

What I dont understand is how be 3 * 2² * 5 makes each of the prime factors have powers with multiples of 3

Could someone help me on this please.
Intern
Joined: 09 Feb 2009
Posts: 17
Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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14 Mar 2010, 14:10
great explanation! thanks!

Bunuel wrote:
gautamsubrahmanyam wrote:
I am not sure I fully understand the explanation.

Method1 asks for a technique to find a value of n divisible by 450.This seems to be a time consuming method
The other method I dont seem to understand.450=3^2*2*5^2
I also understand the statement "for a cube, the powers of the prime factors have to be multiples of 3"

What I dont understand is how be 3 * 2² * 5 makes each of the prime factors have powers with multiples of 3

Could someone help me on this please.

We have $$450y=2*3^2*5^2*y=n^3$$, as $$y$$ and $$n$$ are positive integers, $$y$$ must complete the powers of 2, 3 and 5 so that these powers will be the multiples of 3. Hence the least value of $$y$$ is $$2^2*3*5$$. In this case $$2*3^2*5^2*y=(2*3^2*5^2)*(2^2*3*5)=2^3*3^3*5^3=(2*3*5)^3=n^3$$.

As the least value of $$y$$ must be $$2^2*3*5$$, then $$\frac{y}{2^2*3*5}$$, will equal to 1. Choice B.

Well generally speaking (and little complicating), $$y$$ must be of a form $$y=2^{2+3p}*3^{1+3q}*5^{1+3r}*x^3$$, where $$p$$, $$q$$ and $$r$$ are the integers $$\geq0$$ and $$x$$ ($$x\geq{0}$$), is some other multiple of $$y$$ which also has the power of $$3$$.

For example $$y$$ can be $$y=2^5*3^7*5^{16}*91^3$$, in this case $$(2*3^2*5^2)*(2^5*3^7*5^{16}*91^3)=(2^2*3^3*5^6*91)^3=n^3$$.
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Joined: 14 Dec 2010
Posts: 43
Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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18 Dec 2010, 03:00
viperm5 wrote:
great explanation! thanks!

Bunuel wrote:
gautamsubrahmanyam wrote:
I am not sure I fully understand the explanation.

Method1 asks for a technique to find a value of n divisible by 450.This seems to be a time consuming method
The other method I dont seem to understand.450=3^2*2*5^2
I also understand the statement "for a cube, the powers of the prime factors have to be multiples of 3"

What I dont understand is how be 3 * 2² * 5 makes each of the prime factors have powers with multiples of 3

Could someone help me on this please.

We have $$450y=2*3^2*5^2*y=n^3$$, as $$y$$ and $$n$$ are positive integers, $$y$$ must complete the powers of 2, 3 and 5 so that these powers will be the multiples of 3. Hence the least value of $$y$$ is $$2^2*3*5$$. In this case $$2*3^2*5^2*y=(2*3^2*5^2)*(2^2*3*5)=2^3*3^3*5^3=(2*3*5)^3=n^3$$.

As the least value of $$y$$ must be $$2^2*3*5$$, then $$\frac{y}{2^2*3*5}$$, will equal to 1. Choice B.

Well generally speaking (and little complicating), $$y$$ must be of a form $$y=2^{2+3p}*3^{1+3q}*5^{1+3r}*x^3$$, where $$p$$, $$q$$ and $$r$$ are the integers $$\geq0$$ and $$x$$ ($$x\geq{0}$$), is some other multiple of $$y$$ which also has the power of $$3$$.

For example $$y$$ can be $$y=2^5*3^7*5^{16}*91^3$$, in this case $$(2*3^2*5^2)*(2^5*3^7*5^{16}*91^3)=(2^2*3^3*5^6*91)^3=n^3$$.

As y=2*2*3*5=60,so B as well as C is also correct because II. y/ (3^2 * 2 * 5) is equal to 60/60=1 and that is an integer.

The same way III. y/ 3 * 2 * 5^2 can also be the integer because when we multiply 5*5*5 to y=2*2*3*5 ie 60*125 it provides the III. y/ 3 * 2 * 5^2 with the integer value.

So the answer should be E.

Let me have ur thoughts on the same.
Math Expert
Joined: 02 Sep 2009
Posts: 52336
Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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18 Dec 2010, 03:19
3
Eshika wrote:
As y=2*2*3*5=60,so B as well as C is also correct because II. y/ (3^2 * 2 * 5) is equal to 60/60=1 and that is an integer.

The same way III. y/ 3 * 2 * 5^2 can also be the integer because when we multiply 5*5*5 to y=2*2*3*5 ie 60*125 it provides the III. y/ 3 * 2 * 5^2 with the integer value.

So the answer should be E.

Let me have ur thoughts on the same.

OA for this question is B, not E.

Next, y does not necessarily equal to 2^2*3*5, the least value of $$y$$ equals to $$2^2*3*5$$.

As we are asked which of the following options MUST be an integer then these options must be an integers for the least value of $$y$$:

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, note an integer;

III. y/(3*2*5^2) --> (2^2*3*5)/(3*2*5^2)=2/5, note an integer;

So the answer is B (I only).

Check similar questions:

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number-properties-92562.html?hilit=following%20must%20none#p712584
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division-factor-88388.html#p666722

Hope it helps.
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Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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18 Dec 2010, 03:27
Yeah I got that...Thanks for your quick response.
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Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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14 Apr 2012, 08:01
Bunuel wrote:
gautamsubrahmanyam wrote:
I am not sure I fully understand the explanation.

Method1 asks for a technique to find a value of n divisible by 450.This seems to be a time consuming method
The other method I dont seem to understand.450=3^2*2*5^2
I also understand the statement "for a cube, the powers of the prime factors have to be multiples of 3"

What I dont understand is how be 3 * 2² * 5 makes each of the prime factors have powers with multiples of 3

Could someone help me on this please.

We have $$450y=2*3^2*5^2*y=n^3$$, as $$y$$ and $$n$$ are positive integers, $$y$$ must complete the powers of 2, 3 and 5 so that these powers will be the multiples of 3. Hence the least value of $$y$$ is $$2^2*3*5$$. In this case $$2*3^2*5^2*y=(2*3^2*5^2)*(2^2*3*5)=2^3*3^3*5^3=(2*3*5)^3=n^3$$.

As the least value of $$y$$ must be $$2^2*3*5$$, then $$\frac{y}{2^2*3*5}$$, will equal to 1. Choice B.

Well generally speaking (and little complicating), $$y$$ must be of a form $$y=2^{2+3p}*3^{1+3q}*5^{1+3r}*x^3$$, where $$p$$, $$q$$ and $$r$$ are the integers $$\geq0$$ and $$x$$ ($$x\geq{0}$$), is some other multiple of $$y$$ which also has the power of $$3$$.

For example $$y$$ can be $$y=2^5*3^7*5^{16}*91^3$$, in this case $$(2*3^2*5^2)*(2^5*3^7*5^{16}*91^3)=(2^2*3^3*5^6*91)^3=n^3$$.

Thanks. Now is clear because the question that I had, it was in the form n3 and NOT n^3 so I didn't think how to solve this problem because the to quantities did not balance each other
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Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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12 May 2012, 07:00
2
1
rishi2377 wrote:
If n and y are positive integers and 450y = n^3, which of the following must be an interger

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

OA to follow

A difficult problem which can be easy if we understand

the question states that 450y=n^3 that means 450y must be to the third power of n ,so if a number raised to the power of cube then all numbers must be in 3 no's for example if we take 216 then if we factorize 216 we get 2*2*2*3*3*3 that means in cube power we have each number 3 times so let's go to our problem here we have 3 2's and 3 3's

so 450y must have all numbers in 3 no's so if we factorize 450 we get 450=2*3*3*5*5 as we need all numbers in 3 no's that means we should have 3 2's,3 3's,3 5's so we need extra 2 2's,a 3 and a 5 to make 450y perfect cube
so y must be 2^2*3*5=>4*3*5 =>60 so y must be 60 to make 450y a perfect cube

so if we look at given options

1. y/ (3 * 2^2 * 5) => y/3*4*5 =>y/60 as we know that y is 60 so 60/60=1 which is integer
2. y/ (3^2 * 2 * 5) => y/9*2*5 =>y/90 as we know that y is 60 so 60/90 =>2/3 which is not integer
3. y/ (3 * 2 * 5^2) => y/3*2*25 =>y/150 as we know that y is 60 so 60/150 =>2/5 which is not integer

so we get only 1 as integer so our answer is b

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Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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11 Jul 2012, 11:41
Can someone please explain to me why y cannot simply be 2.

If 450y = n^3, then cannot it be 3x3x5x5x2 and y=2 so that would make 3x3x5x5x2x2 so it ends up 30x30x30 = nxnxn. Then if y=x and the equation works, then y/I, II, or III would not work.

In this case, the answer should be A, None. Please help, I've been racking my brain over why y cannot be 2, but it makes perfect sense.
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Posts: 52336
Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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12 Jul 2012, 01:09
mthasan1 wrote:
Can someone please explain to me why y cannot simply be 2.

If 450y = n^3, then cannot it be 3x3x5x5x2 and y=2 so that would make 3x3x5x5x2x2 so it ends up 30x30x30 = nxnxn. Then if y=x and the equation works, then y/I, II, or III would not work.

In this case, the answer should be A, None. Please help, I've been racking my brain over why y cannot be 2, but it makes perfect sense.

If $$y=2$$ then $$450y=2^2*3^2*5^2=30^2=900$$ not $$30^3$$ as you've written.
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Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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19 Oct 2014, 18:23
rishi2377 wrote:
If n and y are positive integers and 450y = n^3, which of the following must be an interger

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

Hello There,
450 X y = n^3

Factors of 450 = (3^2) X (5^2) X 2
Since, there is a cube on RHS, lets try to make LHS also a cube.
To do this, we need one 3, one 5 and two 2s which will result in 3^3 X 5^3 X 2^3
Now, y = 3 X 5 X 2^2 = 60.

Now verifying with options, only I results in an integer.

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Bharat Bhushan Sunkara.

"You need to sacrifice what you are TODAY, for what you want to be TOMORROW!!"

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Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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26 Dec 2014, 11:01
450y = n^3 means 450y is a cube.

450 = 3² * 2 * 5²

for a cube, the prime factors should be in a set of 3

so Y/3*2²*5
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Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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07 Mar 2015, 09:33
Could someone please explain why 2,3,and 5 have to be in powers that is a multiple of 3? Recognizing that seems to be the crux of the problem. I am not sure why if n^3 that means the prime factors have to be in powers of 3 also. If anyone could provide explanations or links for this rule it would be much appreciated! Thanks!
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Joined: 02 Sep 2009
Posts: 52336
Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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08 Mar 2015, 06:12
khl52 wrote:
Could someone please explain why 2,3,and 5 have to be in powers that is a multiple of 3? Recognizing that seems to be the crux of the problem. I am not sure why if n^3 that means the prime factors have to be in powers of 3 also. If anyone could provide explanations or links for this rule it would be much appreciated! Thanks!

Say $$n = p^2*q^5$$, where p and q are primes, then $$n^3 = (p^2*q^5)^3=p^{(2*3)}*q^{(5*3)}=p^6*q^{15}$$. So, whatever prime factorization n has, n^3 will multiply the powers of the primes by 3.
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Re: If n and y are positive integers and 450y = n^3 which of the following  [#permalink]

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Re: If n and y are positive integers and 450y = n^3 which of the following &nbs [#permalink] 20 Dec 2018, 19:22
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