GMAT PS QOTW : Number Properties

This question is an interesting Problem Solving question from the topic number properties.

How many digits does the product of 4¹² and 5²³ contain?
A. 12
B. 13
C. 23
D. 24
E. 35

The correct answer is choice D.

We can rewrite the given numbers as 2²⁴ * 5²³.
i.e., 2 * 2²³ * 5²³
= 2 * 10²³.
This is a 24 digit number.

Number Properties : Data Sufficiency

Number Properties is an all time favorite with the GMAT test makers. Especially in the DS avtar, number properties questions could be quite potent.

Here is a seemingly innocuous question from Number Properties presented as a DS question.

Question

Is ab positive?
(1) (a+b)^2 < (a-b)^2
(2) a = b

It is an "Is" question. So, the answer has to be a definite YES or a definite NO. It cannot be a MAYBE.

Let us evaluate statement 1.
a^2 + b^2 + 2ab < a^2 + b^2 - 2ab
Simplifying we get, 4ab < 0 or ab < 0.
So, we can convincingly answer that ab is not positive. So, statement 1 is sufficient to answer the question.

The correct answer is either A or D.

Now let us evaluate the statement 2. 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.
As we are not able to conclude if ab is positive or not with statement 2, it is not sufficient.

So, choice A is the correct answer.

Quadratic Equations : Sum of roots, product of roots

Quadratic equations are equations of the form ax² + bx + c = 0.

A quadratic equation has two roots. These roots are found either by factorizing the quadratic equation or by using the formula (-b + root (b² - 4ac))/2a and (-b - root (b² - 4ac))/2a

Here is a typical quadratic equation question

If m and n are the roots of the quadratic equation x² - (2 root 5)x - 2 = 0, the value of m² + n² is:

A. 22
B. 24
C. 32
D. 20
E. 18

Correct Answer is Choice B. 24.

Explanation

m and n are roots of the equation.

We have to find the value of m² + n²

m² + n² = (m + n)² - 2mn

(m + n), the sum of the roots of a quadratic equation of the form ax² + bx + c = 0 is (-b/a)

mn, the product of the roots of the equation = c/a

The sum of the roots of the equation x² - (2 root 5)x - 2 = 0 is (2 root 5).
Product of the roots of the equation = -2.

Hence, (m + n)² - 2mn = (2 root 5)² - 2(-2) = 20 + 4 = 24.

GMAT DS : Geometry, Coordinate Geometry

Here is a data sufficiency question. It is a question on slopes of lines and tests basic concepts about lines in geometry and coordinate geometry.

Question

Are lines p (with slope m) and q (with slope n) perpendicular to each other?
1. m + 2 = n
2. m + n = 0

Correct Answer: Choice C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

Explanatory Answer

If two lines are perpendicular, then the product of the slopes of the two lines will be equal to -1.

In this case, if the product m * n = -1, then the two lines will be perpendicular to each other. If the product is not equal to -1, then they are not perpendicular. We need to assess that conclusively.

Statement 1 m + 2 = n
m could be -1 and n could be 1, in which case the product is -1. Alternatively, m could be 4 and n could be 6 in which case the product is not -1.

As we are not able to conclude using the information in statement 1, it is not sufficient. Choices A and D can be eliminated. We are left with choices B, C or E.

Statement 2 m + n = 0.
m could be -1 and n could be 1 or vice versa. In that case, m * n = -1.
m could be any other number and n could be -m. In that case m * n will not be equal to -1. Hence, statement 2 is also not sufficient. We can eliminate choice B. We are left with choices C or E.

Combining the two statements, we know that m = -n from statement 2. Substituting that in statement 1, we get m + 2 = -m or 2m = -2 or m = -1. Hence, n = 1. Hence, the product m * n = -1.

As the information provided in the two statements is sufficient to answer the question, choice C is the correct answer.

Data Sufficiency : Number Properties

About a third of the questions that appear on the Quantitative section of the GMAT CAT test are data sufficiency (DS) questions. Here is a DS question from Number properties and number theory. Number properties and number theory is a hot favorite when it comes to setting DS questions.

For the uninitiated amongst us, this is how the instructions to a DS question will look like.

Directions

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 -

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

Numbers

All numbers used are real numbers.

Figures

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.

Note: 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

What is the value of b?

1) a = 3
2) (a-3)(b+2)=0

The correct answer is E. i.e., Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Explanatory Answer
Statement 1: It is obvious that statement 1 is not sufficient. Data given does not provide any information about the value of b. So, we can eliminate choices A and D. So we are down to choices B, C or E.

Statement 2: An equation with two variables 'a' and 'b'. Hence, the information provided is not sufficient.

Note do not make the mistake of solving the equation as either (a - 3) = 0 or (b + 2) = 0 and decide that b = -2. What if only (a - 3) = 0 and (b + 2) was not '0'.

Combining the two statements, we know a = 3, so, (a - 3) = 0. But that leaves the question of what (b + 2) or what 'b' is unanswered.

Hence, choice E is the correct answer.

Data sufficiency questions are usually quite tricky. You can ensure success in cracking these questions only with adequate practice. Any serious aspirant should solve about 250 to 300 data sufficiency questions before taking the actual GMAT test.

Permutation Combination and Probability - Sampling with / without ordering

The next parameter on which sampling can be classified is Sampling based on whether Ordering (Arrangement) of the elements selected is considered or not.

In this case too, as in the case of sampling with or without replacement let us look at two examples that will help us learn the concept better.

Example 1. In how many ways can a group of students elect a President and a Vice President from 10 contestants if a person cannot hold more than one post?

It is quite obvious that the president can be elected from the 10 contestants in 10 ways and the vice president can be elected from the remaining 9 students in 9 ways. As the group of students elect a president and a vice president, the total number of ways = 10 * 9 = 90.

Let the contestants be recognized by the letters A to J.
Any one of the A to J could have been elected as the president.
Now, let us say, B was elected the president. Then, anyone of the remaining 9 contestants can be elected as vice president. Say, D was elected vice president.

You will realize that the 90 outcomes include the case of "B" being the president and "D" being the vice president and "B" being the vice president and "D" being the president.

i.e., for B and D being the two contestants who were elected, there are two possibilities - BD or DB and therefore, this is an example of sampling with Ordering

GMAT Tips - Do not skip questions

Here is an insight from the official GMAT® test organizers - GMAC® - The graduate Management Admissions Council.

The GMAC® briefing to the test-preparation community on the transition of the GMAT® exam to new vendors in March - April 2005 had GMAC® Chief Psychometrician Larry Rudner fielded a host of questions about the underpinnings of the computer algorithm that scores the test. He emphasized that candidates must complete every question on the GMAT® exam; skipping a question is not an option, and test takers are best advised to make an educated guess if they are unsure of an answer.

To read the full text of the news, you can visit the following URL
http://www.gmac.com/gmac/VirtualLibrary/Publications/GMNews/2005/MarchApril/GMACTestPrepBriefing.htm

You can also visit the following URL to get an insight into how some students cracked the GMAT and secured admissions from top B Schools.
How I Cracked GMAT?