DS -Number properties

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agni_h1: Junior | Next Rank: 30 Posts; Posts: 14; Joined: Tue Jun 08, 2010 1:19 pm

by agni_h1 » Thu Apr 28, 2011 5:52 am

If x^2 + 5y = 49 . Is y an integer ?

(1) 1<x<4
(2) x^2 is an integer

I selected C. But the correct answer is E.

Please help.

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Source: Beat The GMAT — Data Sufficiency | Thu Apr 28, 2011 5:52 am

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Apr 28, 2011 6:19 am

agni_h1 wrote:If x^2 + 5y = 49 . Is y an integer ?

(1) 1<x<4
(2) x^2 is an integer

I selected C. But the correct answer is E.

Please help.

One way to solve this question is by counter-example.

Statement 1: 1<x<4
Here, we can see that if x=2, then y is an integer. So, when x=2 the answer to the target question is YES.

At this point, we can see that most other values of x (other than x=3) will mean that y is not an integer. However, before we choose a value for x, let's take a quick look at statement 2: x^2 is an integer.

Notice that the first value for x we chose (x=2) is an integer when squared. So, this value of x satisfies statements 1 and 2.
If we can find another value of x that satisfies both statements and that value of x is such that y is not an integer (i.e., the answer to the target question is NO), then we can solve this question very quickly.

Notice that x=root3 satisfies both statements 1 and 2.

So, for statement 1, if x = root3, then y is not an integer.
So statement 1 is INSUFFICIENT

Here's the bonus of choosing numbers that satisfy both statements.

Statement 2: x^2 is an integer
x=2 means that y is an integer.
x=root3 means that y is not an integer.
So statement 2 is INSUFFICIENT

Since we used the same two x values for both statements, we know that the statements combined are
INSUFFICIENT, which means the answer is E

Brent Hanneson - Creator of GMATPrepNow.com

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agni_h1: Junior | Next Rank: 30 Posts; Posts: 14; Joined: Tue Jun 08, 2010 1:19 pm

by agni_h1 » Thu Apr 28, 2011 6:31 am

Thanks. I did think about picking numbers with root. Just another small query if you don't mind.

To prove that statement 1 alone is Insufficient I considered values like 1.5 and 2.5 because it is not mentioned anywhere that x is an integer. Is that the right approach ? These numbers do not satisfy the equation when we take statement 2 into consideration.So, I had assumed C to be the answer.I had not considered picking roots.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Apr 28, 2011 6:55 am

agni_h1 wrote:Thanks. I did think about picking numbers with root. Just another small query if you don't mind.

To prove that statement 1 alone is Insufficient I considered values like 1.5 and 2.5 because it is not mentioned anywhere that x is an integer. Is that the right approach ? These numbers do not satisfy the equation when we take statement 2 into consideration.So, I had assumed C to be the answer.I had not considered picking roots.

Our goal when picking numbers is to find numbers that satisfy a given condition but result in different answers to the target question.

For example, consider the relatively simple question::
If x is greater than 0, is x a prime number?
(1) x is odd

To show that statement 1 is not sufficient by counter-example, we need to find values of x that satisfy statement 1, but yield different answers to the target question.

So, given the condition in statement 1, x could equal 3 in which case the answer to the target question is YES (x is prime)

Now we need to find a value of x that yields an answer of NO.

Well, given the condition in statement 1, x could equal 15 in which case the answer to the target question is NO (x is not prime)

Since statement 1 yields two conflicting answers to the target question, it is not sufficient.

Now let's examine your approach.

While x=1.5 and x=2.5 do satisfy the condition in statement 1 (1<x<4 ), these two values of x don't yield different answers to the target question.

If x=1.5, then y is not an integer
If x=2.5, then y is not an integer
At this point, we cannot be certain that statement 1 is insufficient, since these two values of x do not yield conflicting answers to the target question.

Brent Hanneson - Creator of GMATPrepNow.com

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agni_h1: Junior | Next Rank: 30 Posts; Posts: 14; Joined: Tue Jun 08, 2010 1:19 pm

by agni_h1 » Thu Apr 28, 2011 8:35 am

Thanks for the explanation. That was really useful.

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MAAJ: Master | Next Rank: 500 Posts; Posts: 243; Joined: Sun Jul 12, 2009 7:12 amFollowed by: 2 members; Location: Dominican Republic; Thanked: 31 times; GMAT Score:480

by MAAJ » Fri Apr 29, 2011 6:31 am

You can also factorize to make it simpler.

If xÂ² + 5y = 49 . Is y an integer ?
5y = 49 - xÂ²
5y = (7 + x)(7 - x)
y = [(7 + x)(7 - x)]/5
So if we make one of the parenthesis equal to 5 or a multiple of 5, then y must be an integer.

(1) 1 < x < 4
Integers:
7 - 2 = 5 YES
7 + 3 = 10 YES

Non-integers:
7 - (2/3) = NO
7 - (7/2) = NO

(2) xÂ² is an integer
2Â² = Integer, and 7 +/- 2 is a multiple of 7 - YES
sqrt(2)Â² = Integer, but 7 +/- sqrt(2) is not a multiple of 7 - NO

Combined we can't say that the value of x is integer 2 or 3

"There's a difference between interest and commitment. When you're interested in doing something, you do it only when circumstance permit. When you're committed to something, you accept no excuses, only results."

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