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Bunuel
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If x and y are integers, is xy divisible by 3 ? 24 Jan 2023, 08:29
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C
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65% (hard)

Question Stats:

55% (02:07) correct 45% (01:55) wrong based on 64 sessions

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If x and y are integers, is xy divisible by 3 ?

(1) (x + y)^2 is divisible by 9

(2) (x - y)^2 is divisible by 9.

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C
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gmatophobia
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If x and y are integers, is xy divisible by 3 ? 24 Jan 2023, 13:19
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Bunuel
If x and y are integers, is xy divisible by 3 ?

(1) (x + y)^2 is divisible by 9

(2) (x - y)^2 is divisible by 9.
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Statement 1

(1) (x + y)^2 is divisible by 9

Case 1: \(\frac{(7+2)^2 }{ 9}\)

is xy divisible by 3 ? - No

Case 1: \(\frac{(6+3)^2 }{ 9}\)

is xy divisible by 3 ? - Yes

The statement is not sufficient

Statement 2

(2) (x - y)^2 is divisible by 9

Case 1: \(\frac{(-10 - (-1) )^2 }{ 9}\)

is xy divisible by 3 ? - No

Case 1: \(\frac{(6 - 3)^2 }{ 9}\)

is xy divisible by 3 ? - Yes

The statement is not sufficient

Combined

Adding or substracting two multiples of a number results in a multiple of that number.

\((x + y)^2 - (x-y)^2\) is a multiple of 9

4xy is a multiple of 9

As 4 is not a multiple of 9, therefore xy is a multiple of 9.

Sufficient.

Option C
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If x and y are integers, is xy divisible by 3 ? 07 Mar 2023, 07:46
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gmatophobia,

how do you know to subtract these to find correction solution C?
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If x and y are integers, is xy divisible by 3 ? 07 Mar 2023, 08:07
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gmatophobia,

how do you know to subtract these to find correction solution C?
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We need to find the divisibility of xy.

The information from statements 1 and 2 do not provide any information on xy. We know that

\((x + y)^2 = x^2 + y^2 + 2xy\\
\\
(x - y)^2 = x^2 + y^2 - 2xy\)

So the logical route was to subtract both so as to get 4xy.
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If x and y are integers, is xy divisible by 3 ? 17 Apr 2023, 11:22
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Bunuel
If x and y are integers, is xy divisible by 3 ?

(1) (x + y)^2 is divisible by 9

(2) (x - y)^2 is divisible by 9.
Show more

Official Solution:


If \(x\) and \(y\) are integers, is \(xy\) divisible by 3 ?

(1) \((x + y)^2\) is divisible by 9

Since \(x\) and \(y\) are integers, the given information implies that \(x + y\) is divisible by 3. However, this is not sufficient to determine whether \(xy\) is divisible by 3. For example, if \(x = 1\) and \(y = 2\), the answer is NO, but if \(x = 3\) and \(y = 3\), the answer is YES. This statement is not sufficient.

(2) \((x - y)^2\) is divisible by 9

Since \(x\) and \(y\) are integers, the given information implies that \(x - y\) is divisible by 3. However, this is not sufficient to determine whether \(xy\) is divisible by 3. For example, if \(x = 4\) and \(y = 1\), the answer is NO, but if \(x = 3\) and \(y = 3\), the answer is YES (remember that 0 is divisible by every integer, with the exception of 0 itself). This statement is not sufficient.

(1) + (2) If both \(x + y\) and \(x - y\) are divisible by 3, their sum \((x+y) + (x-y)=2x\) must also be divisible by 3. From the fact that \(2x\) is divisible by 3, we can conclude that \(x\) is divisible by 3. Since \(x\) and \(y\) are integers, it follows that \(xy\) must be divisible by 3. Sufficient.


Answer: C

P.S. Essentially, since both x + y and x - y are multiples of 3, their sum and their difference will also be multiples of 3.


WHEN THE SUM OR THE DIFFERENCE OF NUMBERS IS A MULTIPLE OF AN INTEGER

1. If integers \(a\) and \(b\) are both multiples of some integer \(k>1\) (divisible by \(k\)), then their sum and difference will also be multiples of \(k\) (divisible by \(k\)):

Example: \(a=6\) and \(b=9\), both divisible by 3 ---> \(a+b=15\) and \(a-b=-3\), again both divisible by 3.


2. If out of integers \(a\) and \(b\), one is a multiple of some integer \(k>1\) and the other is not, then their sum and difference will NOT be multiples of \(k\) (divisible by \(k\)):

Example: \(a=6\), divisible by 3, and \(b=5\), not divisible by 3 ---> \(a+b=11\) and \(a-b=1\), neither is divisible by 3.


3. If integers \(a\) and \(b\) both are NOT multiples of some integer \(k>1\) (divisible by \(k\)), then their sum and difference may or may not be multiples of \(k\) (divisible by \(k\)):

Example: \(a=5\) and \(b=4\), neither is divisible by 3 ---> \(a+b=9\), is divisible by 3, and \(a-b=1\), is not divisible by 3;
OR: \(a=6\) and \(b=3\), neither is divisible by 5 ---> \(a+b=9\) and \(a-b=3\), neither is divisible by 5;
OR: \(a=2\) and \(b=2\), neither is divisible by 4 ---> \(a+b=4\) and \(a-b=0\), both are divisible by 4.
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