Is integer R positive?

Hi All,

Certain DS questions are really just a test of your "thoroughness"; you don't have to be amazing at math to get the correct answer, but you DO have to "see" more than just the obvious solution to a given question.

Here, we're told that R is an integer. We're asked if it is POSITIVE. This is a YES/NO question. The phrasing of the prompt gets me thinking that R MIGHT be positive, but it also MIGHT be negative or even 0.

Fact 1: R^3 = R

Most Test Takers look at this and "see" that R = 1 is a solution....but is it the ONLY solution? With a bit more work (or even just 'playing around' with the prompt), you'll see that R= 0 and R = -1 are BOTH possible solutions....

IF...
R = 1 then the answer to the question is YES
R = 0 then the answer to the question is NO
R = -1 then the answer to the question is NO
Fact 1 is INSUFFICIENT

Fact 2: |R| = R

Again, most Test Takers would see that R = 1 is a solution, but R could be ANY positive integer and R = 0 is another solution worth noting.

IF....
R = 1 then the answer to the question is YES
R = 0 then the answer to the question is NO
Fact 2 is INSUFFICIENT

Combined, we know...
R^3 = R
|R| = R

From our prior work, we already can see answers that fit both Facts:
IF....
R = 1 then the answer to the question is YES
R = 0 then the answer to the question is NO
Combined, INSUFFICIENT

Final Answer:
The 'takeaway' from all of this is that, in many DS questions, you have to seek out the potential answers. There's ALWAYS at least one answer - it's just a matter of whether there's more than one or not.

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

This was an excellent question! And, the low accuracy rates suggest that many students got stumped by it, probably because they failed to consider that the modulus function yields non-negative values, and not strictly positive values (in simpler words, the modulus function can also yield a value of zero)

Here is a similar question for you to further practice this very important point:

Is integer x negative?

(A) x is not equal to |x|
(B) x = -|y - 2|, where y is an integer

Please post your solution below. I'll provide the official answer and explanation soon. Till then, Happy Solving and wish you all the best!

Best Regards

Japinder

Hello Japinder, thanks for interesting question )

1) \(x\) can be not equal to \(|x|\) only in case when \(x < 0\). Because if \(x >= 0\) when \(x\) will be equal to \(|x|\)
Sufficient

2) This statement give us two possible variants:
\(x = 0\) and \(y =2\) and then \(0 = -|2 - 2|\)
or \(x = -2\) and \(y = 4\) or \(y = 0\) then \(-2 = -|0-2|\) or \(-2=-|4-2|\)
So \(x\) can be negative or equal to \(0\)
Insufficient

Answer is A

Hi Japinder,

Here goes my approach:

Is integer x negative?

(A) x is not equal to |x|
(B) x = -|y - 2|, where y is an integer


Statement A - if X <> |X| then this implies that x is positive number - Sufficient
Statement B - this introduces another variable 'y' we know nothing about - Insufficient

Hence A is the correct option.

The correct answer is Option A

The question statement just tells us that x is an integer.

This means,
either x is negative
or x = 0
or x is positive

Let's now see if the given statements help us eliminate some of these cases.

(1) x is not equal to |x|

This statement only holds true for x is negative

x = 0 and
x is positive violate Statement 1, and therefore can be ruled out.

Thus, Statement 1 is sufficient to confirm that x is negative.

(2) x = -|y - 2|, where y is an integer

Now, as I pointed out in the post quoted here, the modulus function yields non-negative values, and not strictly positive values (in simpler words, the modulus function can also yield a value of zero)

This means either
|y-2| > 0 (for example, for y = 3, -3 etc.)
and therefore, x = -(positive number) = negative integer

or |y-2| = 0 (for y = 2)
and therefore, x = 0

Thus, using Statement 2 alone, we get that x is either negative or equal to zero. So, it's not sufficient to confirm that x is negative.

Hope you enjoyed working on this question

Japinder

Dear aimtoteach

By going through the solution I've posted just now, you'll see that even though y is just a variable, we can deduce from Statement 2 that x is either negative or zero. Statement 2 is not sufficient only because the possibility of x = 0 also exists.

Suppose, the question had asked 'Is integer x positive?' instead. Then, the very same Statement 2 would have been sufficient to deduce that.

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