Data Sufficiency

ziyuenlau wrote:If p, x, and y are positive integers, y is odd, and p = x^2 + y^2, is x divisible by 4?

(1) When p is divided by 8, the remainder is 5
(2) x - y = 3

Source: GMATPrep
OA=A

Since y is odd, say y = 2n + 1; where n is a non-negative integer

Thus, p = x^2 + y^2 = x^2 + (2n+1)^2 = x^2 + 4n^2 + 4n + 1

p = x^2 + 4n^2 + 4n + 1

Statement 1: When p is divided by 8, the remainder is 5

Say p = 8m + 5; where m is a non-negative integer

Thus, 8m + 5 = x^2 + 4n^2 + 4n + 1

=> x^2 = 8m - 4n^2 - 4n + 4

=> x^2 = 4(2m - n^2 - n + 1)

=> x^2 = 4[2m + (n^2 + n) + 1]

=> x^2 = 4[EVEN + EVEN + ODD]; (n^2 + n) is EVEN whether n is EVEN or ODD.

=> x^2 = 4*ODD

=> x = 2*[sqrt(ODD)]

=> x is not a multiple of 4. Sufficient.

Statement 2: x - y = 3

Since y is odd, say y = 2n + 1; where n is a non-negative integer

=> x - (2n+1) = 3

=> x = 2n + 4

=> x = 2(n +2)

If n is EVEN, x = 2*EVEN = multiple of 4. The answer is YES.

However, If n is ODD, x = 2*ODD = Not a multiple of 4. The answer is NO. Insufficient.

The correct answer: A

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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Hi ziyuenlau,

This question can be solved with a mix of Number Properties and TESTing VALUES.

We're told that P, X and Y are all POSITIVE INTEGERS, that Y is ODD and that P = X^2 + Y^2. We're asked if X is divisible by 4. This is a YES/NO question.

1) When P is divided by 8, the remainder is 5

With this Fact, we know that P must be ODD (since it's 5 more than a multiple of 8... re. 5, 13, 21, 29, 37, 45, 53, 61 etc.). We already know that Y is ODD, so Y^2 will also be an ODD perfect square. Therefore....

P = X^2 + Y^2
Odd = X^2 + Odd

So X^2 MUST be EVEN. We were originally told that X must be an INTEGER, so since X^2 is EVEN, then X is EVEN. Now let's TEST VALUES around possible values of X...

IF...
X = 2 and X^2 = 4, Y = 1 and P = 5, then the answer to the question is NO.

IF....
X = 4 and X^2 = 16, then we run into a problem. Given the possible values for P (re. 5, 13, 21, 29, 37, 45, 53, 61 etc.), and X^2 = 16, Y^2 would then have to be a value in the sequence:

Y^2 = 5, 13, 21, 29, 37, etc.

But none of those are perfect squares. Without a possible corresponding value for Y, X=4 is NOT a possible value here.

IF...
X = 6 and X^2 = 36, Y = 1 and P = 37, then the answer to the question is NO.

IF....
X = 8 and X^2 = 64, then we run into the same problem as before. Given the possible values for P, and X^2 = 64, Y^2 would have to be a value in the sequence:

Y^2 = 5, 13, 21, 29, 37, etc.

This is the same sequence we saw before - and again - none of those are perfect squares. Without a possible corresponding value for Y, X=8 is NOT a possible value here. At this point, we see something of a pattern - positive multiples of 4 do NOT 'fit' what we're told, while even non-multiples-of-4 DO fit. Thus, I'd deduce that X is NOT a multiple of 4 and the answer to the question is ALWAYS NO.
Fact 1 is SUFFICIENT.

2) X - Y = 3

IF....
Y = 1, X = 4 and the answer to the question is YES.
Y = 3, X = 6 and the answer to the question is NO.
Fact 2 is INSUFFICIENT

Final Answer: A

GMAT assassins aren't born, they're made,
Rich

[email protected] wrote: So X^2 MUST be EVEN. We were originally told that X must be an INTEGER, so since X^2 is EVEN, then X is EVEN. Squaring ANY even integer will give you a multiple of 4...

0^2 = 0 = (0)(4)
2^2 = 4 = (1)(4)
4^2 = 16 = (4)(4)
6^2 = 36 = (9)(4)
8^2 = 64 = (16)(4)
Etc.
Thus, X wil ALWAYS be divisible by 4 and the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT

Hey Rich,

In your solution, you proved that x² is divisible by 4, but you concluded that x is divisible by 4.

Cheers,
Brent

Hi Mo2men,

Brent properly defined the error - I lost track of the question while dealing with Fact 1 and instead answered the question "Is X^2 divisible by 4?" I've edited my post accordingly.

After revisiting this prompt, there's a subtle 'issue' with it. GMAT question writers are remarkably good at designing questions that will keep a Test Taker from answering a question correctly if that person makes a mistake. While working through Fact 1, I clearly lost sight of the question that was asked (and answered a different, albeit "similar" question). In the end, I ended up with the correct answer, but I shouldn't have - I made a mistake and I should have gotten this question wrong because of it. This could potentially be described as 'luck', but at higher and higher scoring levels, that type of result isn't supposed to occur. It makes me wonder if this question was in the "active pool" of questions for very long (or if it 'tested out' during the Experimental phase). Regardless, there's a big lesson in all of this: always make sure to answer the question that is ASKED.

GMAT assassins aren't born, they're made,
Rich