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Not getting the explanation provided

by kdn508 » Mon Mar 10, 2014 8:32 am

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If x and y are integers, is x/3 an integer?

STATEMENT 1:

x + 14 = 2y

STATEMENT 2:

2y+1/3 is an integer.

Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
EITHER statement BY ITSELF is sufficient to answer the question.
Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, requiring more data pertaining to the problem.



C


Here's the problem where the explanation doesn't make sense to me!
I see that you can use substitution when you combine the two statements but you get left with x/3 +5. How can you tell thats an integer?



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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Mon Mar 10, 2014 8:57 am

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Hey kdn508,

You need to be careful when transcribing questions, because your post is misleading. The question SHOULD read as follows:
kdn508 wrote:If x and y are integers, is x/3 an integer?
(1) x + 14 = 2y
(2) (2y + 1)/3 is an integer.
Target question: Is x/3 an integer?
IMPORTANT: For x/3 to be an integer, x must be DIVISIBLE by 3. So, the target question is really asking us whether or not x is divisible by 3. So, let's rephrase the question as follows...

REPHRASED target question: Is x divisible by 3?

Aside: We have a free video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

Given: x and y are integers

Statement 1: x + 14 = 2y
There are several values of x and y that satisfy this equation. Here are two:
Case a: x = 6 and y = 10, in which case x IS divisible by 3.
Case b: x = 2 and y = 8, in which case x is NOT divisible by 3.
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: (2y + 1)/3 is an integer.
Since there's no information about x, we cannot answer the target question with certainty.
So, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x + 14 = 2y

Statement 2 tells us that (2y + 1)/3 is an integer.
This means that (2y + 1) is divisible by 3
If we take this information and replace 2y with x + 14 [from statement 1], we get: (x + 14 + 1) is divisible by 3
Simplify to get: (x + 15) is divisible by 3
Since 15 is divisible by 3, we can conclude that x MUST also be divisible by 3
Since we can answer the REPHRASED target question with certainty, the combined statements are SUFFICIENT

Answer = C


Aside: To conclude that x is divisible by 3, I used a nice rule that says, "If k is divisible by d, and q is divisible by d, then (k+q) is divisible by d"
A corollary to this rule says, "If k is divisible by d, and (k+q) is divisible by d, then q is divisible by d."

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by kdn508 » Mon Mar 10, 2014 10:32 am

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Hi Brent!

Sorry I was in a rush (on a work break) but thanks for clarifying!
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by [email protected] » Mon Mar 10, 2014 11:50 am

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Hi kdn508,

Brent has provided a great explanation for how TESTing Values can be used to solve this problem. Here's another approach that involves Number Properties.

We're told that X and Y are INTEGERS. We're asked if X/3 is an integer. This is a YES/NO question.

We can "rewrite" the question though....It's ultimately asking "Is X a multiple of 3?"

Fact 1: X + 14 = 2Y

There are several Number Properties involved in this statement...

2Y = 2(Integer), so 2Y is EVEN

X + Even = Even

X = Even - Even, so X is EVEN

If X = 2, then the answer to the question is NO
If X = 6, then the answer to the question is YES
Fact 1 is INSUFFICIENT

Fact 2: (2Y+1)/3 = Integer

This tells us NOTHING about X, so it's INSUFFICIENT.

It DOES tell us that 2Y + 1 = 3(Integer)

So 2Y = 3(Integer) - 1

Combined, we know....

X + 14 = 2Y
2Y = 3(Integer) - 1

So...
X + 14 = 3(Integer) - 1
X = 3(Integer) - 15
X = 3(Integer - 5)

X MUST be a multiple of 3
The answer to the question is ALWAYS YES
Combined, SUFFICIENT

Final Answer: C

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by GMATGuruNY » Mon Mar 10, 2014 12:54 pm

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kdn508 wrote:If x and y are integers, is x/3 an integer?

STATEMENT 1:

x + 14 = 2y

STATEMENT 2:

(2y+1)/3 is an integer.
Question stem, rephrased: Is x divisible by 3?

Statement 1: x+14 = 2y
Since the question stem asks about x, rephrase statement 1 in terms of x:
x = 2y - 14
x = 2(y-7).

x will be a divisible by 3 if y-7 is divisible by 3.
Case 1: y-7 = 3
In this case, x = 2(y-7) = 2*3 = 6, which IS divisible by 3.

x will NOT be divisible by 3 if y-7 is NOT divisible by 3.
Case 2: y-7 = 1
In this case, x = 2(y-7) = 2*1 = 2, which is NOT divisible by 3.
INSUFFICIENT.

Statement 2: (2y+1)/3 is an integer
In other words, 2y+1 is a multiple of 3.
No information about x.
INSUFFICIENT.

Statements combined:
Statement 2 implies that 2y+1 is a multiple of 3.
Make a list of options and simplify:
2y+1 = 3, 6, 9, 12, 15, 18, 21...
Subtracting 1 from every value, we get:
2y = 2, 5, 8, 11, 14, 17, 20
Dividing every value by 2, we get:
y = 1, 5/2, 4, 11/2, 7, 17/2, 10...
Since y must be an integer, only the values in red are viable:
y = 1, 4, 7, 10...

Statement 1 implies that x will be divisible by 3 if y-7 is divisible by 3.
Testing our list of options for y, we get:
If y=1, then y-7 = -6.
If y=4, then y-7 = -3.
If y=7, then y-y = 0.
If y=10, then y-7 = 3.
In every case, y-7 is divisible by 3.
(Note that 0 is divisible by every integer, including 3.)
SUFFICIENT.

The correct answer is C.
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by Jeff@TargetTestPrep » Mon Dec 18, 2017 9:13 am

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kdn508 wrote:If x and y are integers, is x/3 an integer?

1) x + 14 = 2y

2) (2y+1)/3 is an integer
We are given that x and y are integers and need to determine whether x/3 is an integer.

Statement One Alone:

x + 14 = 2y

The information in statement one is not sufficient to answer the question. For example, if x = 4 (and y = 9), x is not divisible by 3; however if x = 6 (and y = 10), then x is divisible by 3. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

(2y+1)/3 is an integer.

We see that y can be values such as 1, 4, 7, etc.

Thus, we see that y = 3k - 2, where k is an integer.

However, since we do not know anything about x, statement two alone is not sufficient to answer the question.

Statements One and Two Together:

We can substitute 3k - 2 for y in statement one and we have:

x + 14 = 2(3k - 2)

x + 14 = 6k - 4

x = 6k - 18

x = 6(k - 3)

Since x is a multiple of 6, x/3 will always be an integer.

Answer: C

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