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# If x and y are positive integers, is the product xy divisible by 9?

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Math Expert
Joined: 02 Sep 2009
Posts: 58427
If x and y are positive integers, is the product xy divisible by 9?  [#permalink]

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01 Oct 2015, 01:47
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If x and y are positive integers, is the product xy divisible by 9?

(1) The product xy is divisible by 6.
(2) x and y are perfect squares.

Kudos for a correct solution.

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Re: If x and y are positive integers, is the product xy divisible by 9?  [#permalink]

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01 Oct 2015, 06:59
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Bunuel wrote:
If x and y are positive integers, is the product xy divisible by 9?

(1) The product xy is divisible by 6.
(2) x and y are perfect squares.

Kudos for a correct solution.

Statement 1: The product xy is divisible by 6.
i.e. one of x and y is certainly divisible by 3 but we can't say about 9 as multiple of 6 may or may not be a multiple of 9. Hence,
NOT SUFFICIENT

Statement 2: x and y are perfect squares.
We have no information whether any one of x and y is a multiple of 3 or not but if anyone were a multiple of 3 then the number must have been a multiple of $$3^2$$ as well. Hence,
NOT SUFFICIENT

Combining the twos statements:
One of x and y is a multiple of 3 (From statement 1)
x and y are perfect squares (From statement 2)
i.e. the number multiple of 3 and a perfect square MUST be a multiple of $$3^2$$ i.e. 9
SUFFICIENT

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Re: If x and y are positive integers, is the product xy divisible by 9?  [#permalink]

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05 Oct 2015, 06:25
Bunuel wrote:
If x and y are positive integers, is the product xy divisible by 9?

(1) The product xy is divisible by 6.
(2) x and y are perfect squares.

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

Question Type: Yes/No The question asks whether xy is divisible by 9. Another way to say this is “Is xy a multiple of 9?” and, further breaking down the question, “Does xy have prime factors that include 3 • 3?” By manipulating the question using conceptual understanding you are often in a stronger position to evaluate the answer choices. Now you know that if the information can guarantee that 32 is a factor of xy then that information is sufficient.

Given information from the question stem: x and y are positive integers.

Statement 1: The product xy is divisible by 6. This tells you that 3 and 2 are factors of xy. However this statement is not sufficient alone. Conceptual understanding will tell you that if 3 is a factor of xy then it is possible that 9 is also a factor of xy. However it is equally possible that 9 is not a factor of xy. If you are in doubt you can use numbers and Play Devil’s Advocate. xy could equal 6 or 12, both of which are divisible by 6 as the statement requires and are not multiples of 9. This would yield an answer of “no.” xy could also equal 36, which is a multiple of 6 and also of 9. This would yield the answer of “yes.” Since yes and no answers are both possible this statement is not consistent and is therefore not sufficient. Eliminate choices A and D.

Statement 2: x and y are perfect squares. Conceptually this is clearly not sufficient. While each of the number must be a perfect square this statement does not guarantee that the result will even be a multiple of 3, much less of 9. Eliminate choice B.

Together: Statement 1 tells you that xy must include the prime factors of 2 and 3 (in order to be a multiple of 6). Since Statement 2 requires x and y to be perfect squares the only way to have a 2 and a 3 as prime factors of x and y is to have 2^2 and 3^2 as factors of x and y. In fact, the smallest values for x and y are 4 and 9. So the answer to the question “Is xy divisible by 9?” is “yes.” Together the statements yield one consistent answer and the correct answer is C.
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If x and y are positive integers, is the product xy divisible by 9?  [#permalink]

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05 Oct 2015, 08:55
2
Bunuel wrote:
If x and y are positive integers, is the product xy divisible by 9?

(1) The product xy is divisible by 6.
(2) x and y are perfect squares.

Kudos for a correct solution.

Target question: Is the product xy divisible by 9?

Given: x and y are positive integers

Statement 1: The product xy is divisible by 6
This statement doesn't FEEL sufficient, so I'll TEST some values.
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 3 and y = 6, in which case xy IS divisible by 9
Case b: x = 2 and y = 3, in which case xy is NOT divisible by 9
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Aside: For more on this idea of plugging in values when a statement doesn't feel sufficient, you can read my article: http://www.gmatprepnow.com/articles/dat ... lug-values

Statement 2: The largest and smallest of the numbers are odd
This statement doesn't FEEL sufficient either, so I'll TEST some values.
Case a: x = 1 and y = 9, in which case xy IS divisible by 9
Case b: x = 1 and y = 4, in which case xy is NOT divisible by 9
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that either p or q is divisible by 3
Statement 2 tells us that both p and q are perfect squares. So, whichever number is divisible by 3 must ALSO be divisible by 9. Since one of the values (x or y) must be divisible by 9, we can be CERTAIN that the product xy is divisible by 9

Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent
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If x and y are positive integers, is the product xy divisible by 9?  [#permalink]

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16 Mar 2016, 09:44
1
If $$x$$ and $$y$$ are positive integers, is the product $$xy$$ divisible by 9?

(1) The product $$xy$$ is divisible by 6.

so product $$xy$$ can have values raging $${6, 12, 18, 24, 30,36.............. }$$

out of all these values only 18, 36,...........are divisible by 9 ........................thereby giving yes to the above question

and rest are divisible by only 3 not 9...........................................................thereby giving no to the above question

Since we have values that give both yes and no answers , The statement 1 is insufficient.

(2) x and y are perfect squares.

Let $$x=a^2$$ and $$y=b^2$$ where a, b are integers.

so $$x$$ and $$y$$ can be 1, 4, 9, 16.....................

$$xy$$ can be 4, 9, 16, 36, 64.............

Even here we have values that give both yes and no answers, The statement 2 is insufficient.

Combining both the statements (1) & (2)

The product $$xy$$ is divisible by 6
$$x$$ and $$y$$ are perfect squares.

$$x$$ and $$y$$ are perfect squares according statement 2 so product $$xy=(ab)^2$$

all the squares in the set {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, ..............144.............. } are taken into consideration.

i.e., 36, 144...........all of which are divisible by 9.....i.e, definitely YES

Therefore data is sufficient here and Ans is C.
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Re: If x and y are positive integers, is the product xy divisible by 9?  [#permalink]

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25 Jan 2018, 04:14
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Re: If x and y are positive integers, is the product xy divisible by 9?   [#permalink] 25 Jan 2018, 04:14
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