This GMAT quant practice question is a data sufficiency question from Number Theory. Concept: properties of positive and negative numbers.

This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- EACH statement ALONE is sufficient to answer the question asked.
- Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

All numbers used are real numbers.

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2)

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

In data sufficiency problems that ask for the value of a quantity, the data given in the statement are sufficient only when it is possible to determine exactly one numerical value for the quantity.

Question 28: Is ab positive?

- (a + b)
^{2}< (a-b)^{2} - a = b

**What kind of an answer will the question fetch?**

The question is an **"Is"** question. Answer to an "is" questions is either YES or NO.

**When is the data sufficient?**

The data is sufficient if we are able to get a DEFINITE YES or a DEFINITE NO from the information given in the statements.

Evaluate Statement (1) ALONE: (a + b)

Expanding both sides of the inequality, we get a^{2} + b^{2} + 2ab < a^{2} + b^{2} - 2ab

Simplifying we get, 4ab < 0 or ab < 0.

So, we can conclude that ab is not positive. We have got a definite NO as the answer.

**Statement 1 ALONE is sufficient.**

Eliminate choices B, C and E. Choices narrow down to A or D.

Evaluate Statement (2) ALONE: a = b

This is actually the statement that could trick you.

a = b.

So, either both a and b or positive or both a and b are negative. __In either case ab is positive__.

We will certainly be "tempted" to decide that statement 2 is also sufficient.

**The catch is that**, both a and b could be 0. In that case __ab = 0, which is not positive__.

If 'a' and 'b' are either both positive or both negative, the answer is yes. If both are 0, the answer is no.

As we are not able to conclude whether ab is positive with statement 2, it is not sufficient.

**Statement 2 ALONE is NOT sufficient.**

Eliminate choice D.

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1. Number Systems | Types of Numbers | Chart

2. Number Properties | Rational & Irrational Numbers

3. GMAT Number Properties | Indices & Rule of Exponents

4. Number Systems | Surds & Conjugates

5. Number Properties | Tests of divisibility

6. Number Properties | How to check whether a number is prime?

7. Number Properties | How to prime factorize a number?

8. GMAT Number Theory | Prime factorization | Properties of squares & cubes

9. Number Properties | What is HCF? | How to find HCF?

10. Number Properties | What is LCM? | How to find LCM?

11. 3 important properties of LCM & HCF | LCM & HCF of fractions

12. Number Properties | When to use LCM and HCF?

13. Number Theory | How to find number of factors?

14. Number Theory | Number of ways to express as a product of 2 factors

15. Number Theory | Sum of all factors of a number

16. Number Theory | Product of all factors of a number

17. Number Theory | Remainders of sum & product

18. Number Theory | Remainder of dividing x^{n} by 'd'

19. Polynomials | Remainder when a monomial divides it

20. Number Theory | Highest power of a prime that divides factorial of 'n'

21. Number Theory | Highest power of a composite number that divides factorial of 'n'

22. Number Theory | Number of trailing zeroes in a number

23. Number Theory | Unit digit of higher powers of numbers

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