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DS Is X^2 * Y^4 an integer divisible by 9

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by mehravikas » Wed Dec 23, 2009 8:54 pm
Answer should be D

1. Statement 1 is sufficient itself
2. Statement 2 can be written as X*Y * X*Y^3 - We know that X*Y is divisible by 9 therefore it is sufficient
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by Stuart@KaplanGMAT » Thu Dec 24, 2009 12:50 am
mmslf75 wrote:Is X^2 * Y^4 an integer divisible by 9 ?

1. X is an integer divisible by 3
2. X*Y is an integer divisble by 9


OA : E

Source : GMATClub

Query :

Doesnt St1 all by itself anwer the question ??
The OA is correct.

One thing we must NEVER do in DS - make assumptions; nowhere in this question does it say that X and Y are integers.

For statement 1, even if X is a multiple of 3, Y could be .0000001, giving us a "no" answer.

We could also pick x=3 and y=1 to get a "yes" answer.

(2) is a bit more interesting, but also insufficient. We can rewrite the original as:

XY * XY * Y * Y

and we can see that even if XY is a multiple of 9, the whole thing doesn't need to be a multiple of 9. For example, we could choose:

x = 9/pi
y = pi

so that xy = 9, and we'd get:

9 * 9 * pi * pi which certainly isn't a multiple of 9.

So, (2) is also insufficient alone.

How about together? We know that X is a multiple of 3 and XY is a multiple of 9.

We can certainly get a yes answer if we pick x=3 and y=3. So the question is, can we get a no answer?

We can no longer pick 9/pi for x, since x must be a multiple of 3.

Well, what if we pick x = 6 and y = 1.5.

6 is a multiple of 3, so we satisfy (1).
6*1.5 = 9 which is a multiple of 9, so we satisfy (2).

9*9*1.5*1.5 isn't an integer, so we can also get a no answer.

Therefore, choose (E).
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by papgust » Thu Dec 24, 2009 1:08 am
Stuart, i was particularly confused with the statement "Is X^2 * Y^4 an integer divisible by 9?" which made me think that statement 1 could be sufficient to answer.
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by Stuart@KaplanGMAT » Thu Dec 24, 2009 1:33 am
papgust wrote:Stuart, i was particularly confused with the statement "Is X^2 * Y^4 an integer divisible by 9?" which made me think that statement 1 could be sufficient to answer.
Yah, the "an integer" part is irrelevant; only integers are divisible by integers. For some reason a lot of questions have that kind of redundancy built in (another common example is that a question will describe a number as an "even integer", when "even" already implies "integer", at least for GMAT level math).

You also have to interpret the question correctly. "Is X^2 * Y^4 an integer divisible by 9?" doesn't imply that X^2 * Y^4 is an integer, just as "is my car a 4 year old Ford?" doesn't imply that my car is 4 years old.
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by papgust » Thu Dec 24, 2009 2:41 am
That's true Stuart.. Even i faced a question few mins back in DS folder saying "an even integer". These kinds of redundant phrases would actually confuse students (who are very new to math) making them think that integers and non-integers can both be even (which in reality is absurd!). No doubt that this question is poorly worded!
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by mmslf75 » Thu Dec 24, 2009 9:31 am
Stuart Kovinsky wrote:
papgust wrote:Stuart, i was particularly confused with the statement "Is X^2 * Y^4 an integer divisible by 9?" which made me think that statement 1 could be sufficient to answer.
Yah, the "an integer" part is irrelevant; only integers are divisible by integers. For some reason a lot of questions have that kind of redundancy built in (another common example is that a question will describe a number as an "even integer", when "even" already implies "integer", at least for GMAT level math).

You also have to interpret the question correctly. "Is X^2 * Y^4 an integer divisible by 9?" doesn't imply that X^2 * Y^4 is an integer, just as "is my car a 4 year old Ford?" doesn't imply that my car is 4 years old.
So it's kinda dangerous when playing with these kind of questions,,

I hope on real gMAT we do not get such questions...!
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by mmslf75 » Thu Dec 24, 2009 9:34 am
Stuart Kovinsky wrote:
mmslf75 wrote:Is X^2 * Y^4 an integer divisible by 9 ?

1. X is an integer divisible by 3
2. X*Y is an integer divisble by 9


OA : E

Source : GMATClub

Query :

Doesnt St1 all by itself anwer the question ??
The OA is correct.

One thing we must NEVER do in DS - make assumptions; nowhere in this question does it say that X and Y are integers.

For statement 1, even if X is a multiple of 3, Y could be .0000001, giving us a "no" answer.

We could also pick x=3 and y=1 to get a "yes" answer.

(2) is a bit more interesting, but also insufficient. We can rewrite the original as:

XY * XY * Y * Y

and we can see that even if XY is a multiple of 9, the whole thing doesn't need to be a multiple of 9. For example, we could choose:

x = 9/pi
y = pi

so that xy = 9, and we'd get:

9 * 9 * pi * pi which certainly isn't a multiple of 9.

So, (2) is also insufficient alone.

How about together? We know that X is a multiple of 3 and XY is a multiple of 9.

We can certainly get a yes answer if we pick x=3 and y=3. So the question is, can we get a no answer?

We can no longer pick 9/pi for x, since x must be a multiple of 3.

Well, what if we pick x = 6 and y = 1.5.

6 is a multiple of 3, so we satisfy (1).
6*1.5 = 9 which is a multiple of 9, so we satisfy (2).

9*9*1.5*1.5 isn't an integer, so we can also get a no answer.

Therefore, choose (E).
Stuart,

U mean to say :

X is Multiple of 3 but since Y could be .0000001 we have a result, that's not INTEGER but is divisible by 9

Dont u think answer will be YES" then ???
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by mehravikas » Thu Dec 24, 2009 1:40 pm
If X is an integer divisible by 3 then X^2 would be divisible by 9 - Seems to be sufficient
If X*Y is divisible by 9 we do not need to worry about X^2 * Y^4 because it will always be divisible by 9

I am not able to understand what is wrong in my assumption.
mehravikas wrote:Answer should be D

1. Statement 1 is sufficient itself
2. Statement 2 can be written as X*Y * X*Y^3 - We know that X*Y is divisible by 9 therefore it is sufficient
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by rohan_vus » Thu Dec 24, 2009 1:49 pm
X and Y are not mentioned as integers so they could be any number..Your point is valid only if X and Y could have been integers

1. X is an integer divisible by 3
Take X = 3 and Y = 1.5 --> x^2*Y^4 ----> clearly not divisible by 9 even though X^2 alone is divisble by 9... NOT SUFFICIENT

2. X*Y is an integer divisble by 9 --> Assume X = 9 and Y = 1/9...X*Y is an integer but X^2*Y^4 = 81* (1/9^4)== (1/81^2)..a fraction thus NOT SUFFICIENT

Even if you combine X and Y..no success as X = 9 case we already considered for 2nd stmnt , which any how suffices case of stmnt 1 of X being multiple of 3..

So (E) ...
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by mehravikas » Thu Dec 24, 2009 2:08 pm
rohan_vus wrote:X and Y are not mentioned as integers so they could be any number..Your point is valid only if X and Y could have been integers

1. X is an integer divisible by 3
Take X = 3 and Y = 1.5 --> x^2*Y^4 ----> clearly not divisible by 9 even though X^2 alone is divisble by 9... NOT SUFFICIENT
- Why take the whole product 3 * 3 * 1.5^4 - isnt it divisible by 9?

2. X*Y is an integer divisble by 9 --> Assume X = 9 and Y = 1/9...X*Y is an integer but X^2*Y^4 = 81* (1/9^4)== (1/81^2)..a fraction thus NOT SUFFICIENT - your example is wrong it x*y is an integer divisible by 9 but in your example x*y = 1 which is not divisible by 9

Even if you combine X and Y..no success as X = 9 case we already considered for 2nd stmnt , which any how suffices case of stmnt 1 of X being multiple of 3..

So (E) ...
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by rohan_vus » Thu Dec 24, 2009 2:34 pm
X and Y are not mentioned as integers so they could be any number..Your point is valid only if X and Y could have been integers

1. X is an integer divisible by 3
Take X = 3 and Y = 1.5 --> x^2*Y^4 ----> clearly not divisible by 9 even though X^2 alone is divisble by 9... NOT SUFFICIENT
- Why take the whole product 3 * 3 * 1.5^4 - isnt it divisible by 9?
--because if you do that total is what ..its 45.5625---> a non integer.. the decimal portion will stand out as remainder when u divide by 9....
2. X*Y is an integer divisble by 9 --> Assume X = 9 and Y = 1/9...X*Y is an integer but X^2*Y^4 = 81* (1/9^4)== (1/81^2)..a fraction thus NOT SUFFICIENT - your example is wrong it x*y is an integer divisible by 9 but in your example x*y = 1 which is not divisible by 9

a mistake indeed coz of hurry , but doesnt matter ... ok X*Y = 9 lets say,., so what -- X = 6 and Y = 3/2 ..X*Y = 9 .. but X^2*Y^4 = 729/4 ...a clear non integer..even if divide a non integer by 9 its the decimal part which stand out to deny the divisibility criteria... Likeiwse there are n number of ways to illustrate this one

Even if you combine X and Y..no success as X = 9 case we already considered for 2nd stmnt , which any how suffices case of stmnt 1 of X being multiple of 3..


So (E) ...[/quote][/quote]
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by mehravikas » Thu Dec 24, 2009 5:29 pm
the content of the questions seems to be ambiguous. You are first calculating the product of X and Y and then dividing the total product by 9.

I am breaking the product as X * X * Y * Y * Y * Y and dividing it by 9.

So the final equation is - (X * X * Y * Y * Y * Y) / 9

If you divide by 9 before multiplying the numerator you can say that the equation is divisible by 9

I agree that with your approach the answer would be E, but consider my approach and check the answer.

I may be wrong but I think it could be solved the way I have done.
rohan_vus wrote:X and Y are not mentioned as integers so they could be any number..Your point is valid only if X and Y could have been integers

1. X is an integer divisible by 3
Take X = 3 and Y = 1.5 --> x^2*Y^4 ----> clearly not divisible by 9 even though X^2 alone is divisble by 9... NOT SUFFICIENT
- Why take the whole product 3 * 3 * 1.5^4 - isnt it divisible by 9?
--because if you do that total is what ..its 45.5625---> a non integer.. the decimal portion will stand out as remainder when u divide by 9....
2. X*Y is an integer divisble by 9 --> Assume X = 9 and Y = 1/9...X*Y is an integer but X^2*Y^4 = 81* (1/9^4)== (1/81^2)..a fraction thus NOT SUFFICIENT - your example is wrong it x*y is an integer divisible by 9 but in your example x*y = 1 which is not divisible by 9

a mistake indeed coz of hurry , but doesnt matter ... ok X*Y = 9 lets say,., so what -- X = 6 and Y = 3/2 ..X*Y = 9 .. but X^2*Y^4 = 729/4 ...a clear non integer..even if divide a non integer by 9 its the decimal part which stand out to deny the divisibility criteria... Likeiwse there are n number of ways to illustrate this one

Even if you combine X and Y..no success as X = 9 case we already considered for 2nd stmnt , which any how suffices case of stmnt 1 of X being multiple of 3..


So (E) ...
[/quote][/quote]
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by rohan_vus » Thu Dec 24, 2009 6:16 pm
In either case , what makes the X^2*Y^4 divisble by 9.. Its plain simple ..it means (X * X * Y * Y * Y * Y) / 9 is an integer .. If its not an integer then 9 cant be said to be evenly dividing X^2*Y^4..



When you say X*Y is divisible by 9 that means (X*Y)^2 is dividble by 9 again ( as its divisble by 81 now) but remaing Y^2 is still needed to make complete product X^2*Y^4... Its basically value of Y which can destroy the thing here...

X*Y being an integer and X being an integer still cant ensure Y is an integer... as X = 18 but Y = 4.5 still X*Y is an integer...point is X*Y u cant tell what Y could be ...

But consider this,
a) X = any integer

b)X+Y = any integer OR
X - Y = any integer
Then Y is definitely an integer using statement a and b...

All its boils down to this... if any number when divided by any number comes out in decimals , the divisor cant be considered as factor ..or in other words , it can be said to divide the the other one evenly...
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by Stuart@KaplanGMAT » Thu Dec 24, 2009 6:22 pm
mehravikas wrote:the content of the questions seems to be ambiguous. You are first calculating the product of X and Y and then dividing the total product by 9.

I am breaking the product as X * X * Y * Y * Y * Y and dividing it by 9.

So the final equation is - (X * X * Y * Y * Y * Y) / 9

If you divide by 9 before multiplying the numerator you can say that the equation is divisible by 9

I agree that with your approach the answer would be E, but consider my approach and check the answer.

I may be wrong but I think it could be solved the way I have done.
I'm not sure how you've solved at all.

You can certainly rewrite the question as:

is (X*X*Y*Y*Y*Y) divisible by 9?

Your solution implies that anything multiplied by 9 is a multiple of 9; that's certainly true if we're multiplying by an integer, but not necessarily true if we're multiplying by a fraction.

Let's simplify the question:

is XY divisible by 9?

(1) x = 9

this might seem sufficient at first glance, but what if y = 1/2? In that case, then xy = 4.5, which is certainly not a multiple of 9.

They key to this question is to realize that Y doesn't have to be an integer; for all integer values of Y we get a "yes" answer (from (1) or (2) alone), but for most non-integer values of Y we get a "no" answer, even after combining the statements.
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