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# If x and y are positive integers, is x^16 - y^8 + 345y^2

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Intern
Joined: 30 May 2008
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If x and y are positive integers, is x^16 - y^8 + 345y^2 [#permalink]

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13 Jul 2009, 02:23
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If x and y are positive integers, is x^16 - y^8 + 345y^2 divisible by 15?

1. x is a multiple of 25, and y is a multiple of 20
2. y = x^2

Statement (1) by itself is not sufficient. For a number to be a multiple of 15, it has to be divisible by both 3 and 5. As both 25 and 20 are multiples of 5, we need to check if $$x^{16} - y^8$$ is divisible by 3. $$x^{16} - y^8$$ can be rewritten as $$(x^2 - y)(x^2 + y)(x^4 + y^2)(x^8 + y^4)$$ . We only need to check if expressions from first two parentheses are divisible by 3. We'll pick 50 and 60 for that purpose:
$$50^2 - 60 = 2500 - 60 = 2440$$
$$50^2 + 60 = 2500 + 60 = 2560$$

Neither of these numbers is divisible by 3. Therefore, $$x^{16} - y^8$$ isn't divisible by 3 (neither by 15) either.

Based on the given explaination, I've the following questions:

1) How do we actually get x^16 - y^8 to (x^2 - y)(x^2 + y)(x^4 + y^2)(x^8 + y^4)?
2) Why do we only need to check if the first two expressions in parentheses are divisible by 3?

Thanks!

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Director
Joined: 03 Jun 2009
Posts: 772
Location: New Delhi
WE 1: 5.5 yrs in IT
Re: gmatClub test 2, Qu2. [#permalink]

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13 Jul 2009, 02:44
sudimba wrote:
1) How do we actually get x^16 - y^8 to (x^2 - y)(x^2 + y)(x^4 + y^2)(x^8 + y^4)?

Solving backwards:
$$(x^2 - y)(x^2 + y)(x^4 + y^2)(x^8 + y^4)$$
...merging the 1st two using the formula $$(a+b)(a-b) = (a^2 - b^2)$$
$$= (x^4 - y^2)(x^4 + y^2)(x^8 + y^4)$$
$$= (x^8 - y^4)(x^8 + y^4)$$
$$=(x^16 - y^8)$$

sudimba wrote:
2) Why do we only need to check if the first two expressions in parentheses are divisible

Not sure
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Re: gmatClub test 2, Qu2. [#permalink]

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13 Jul 2009, 07:18
Curious to know the answer to this question. As I generally shy away from factoring complex expressions such as x^16 - y^8, my approach (which may very well be wrong) is more rudimentary:

For x^16 - y^8 + 345y^2to be divisible by 15, each of the terms in the expression must be divisible by 15.

Statement 1 is insufficient. Both x and y could be multiples of 15 (if, for example, x=75 and y=60), but we don't have enough information to know whether this is the case.

Statement 2 is sufficient. If y=x^2, we can simplify the expression as follows:

x^16 - (x^2)^8 + 345y^2
= x^16 - x^16 + 345y^2
= 345y^2

We know that 345y^2 is always divisible by 15 because 345 is divisible by both 3 and 5 and multiplying any number by a multiple of 15 yields a result that is divisible by 15. Thus, x^16 - y^8 + 345y^2 is divisible by 15 where y=x^2. The answer is B.

Last edited by carriedinterest on 13 Jul 2009, 08:02, edited 2 times in total.
Director
Joined: 03 Jun 2009
Posts: 772
Location: New Delhi
WE 1: 5.5 yrs in IT
Re: gmatClub test 2, Qu2. [#permalink]

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13 Jul 2009, 22:24
carriedinterest wrote:
For x^16 - y^8 + 345y^2to be divisible by 15, each of the terms in the expression must be divisible by 15.

Sorry to contradict you. But I don't agree on this.

Since 345y^2 is already divisible by 15, I would put it this way that for x^16 - y^8 + 345y^2 to be divisible by 15, we will need to prove (x^16 - y^8) is also divisible by 15.

Note: each of these terms may not be divisible by 15 separately, but may be divisible jointly.

e.g x = 25 and y = 20

Now from one of the previous calculations (posted in previous post)
(x^2 + y) is factor of (x^16 - y^8)

(x^2 + y) = 25^2 + 20 = 625 + 20 = 645, which is divisible by 15

This is just one case. If we can prove that for some other case, its not divisible, we can say option A is insufficient.

--== Message from GMAT Club Team ==--

This is not a quality discussion. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.

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Re: gmatClub test 2, Qu2.   [#permalink] 13 Jul 2009, 22:24
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