GMAT DS : Number Theory

Number Theory is a big hit with the GMAT test setters, especially when setting Data Sufficiency Questions.

Here is a GMAT DS practice question from the Number Theory topic.

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.

Question
When Y is divided by 2, is the remainder 1?
1. (-1)^(Y+2) = -1
2. Y is prime

The given question is an "Is" question. "IS" questions have to answered with an unswerving YES or NO. If your answer to this question is SOMETIMES YES and SOMETIMES NO or in other words MAYBE, then you have not answered the question.

Let us evaluate statement 1.

(-1)^(Y+2) = -1.
(-1)ODD NUMBER = -1

Therefore, Y + 2 is an odd number.
Hence, Y has to be an odd number.

So, when Y is divided by 2, the remainder is 1.

Statement 1 is sufficient.
The answer is either choice (A) or choice (D).


Now let us evaluate the statement 2.

Y is prime

Y could be '2' which is an even number.
So, when Y is divided by 2, the remainder is '0'.

All other prime numbers are odd numbers.
So, when Y is divided by 2, the remainder is '1'.

We cannot conclude is Y is 2 or other prime numbers.

As we are not able to conclude if Y is an even number or an odd number with statement 2, it is not sufficient.

Hence, answer is choice (A ).

Number of diagonals in convex polygon : GMAT

Any n-sided convex polygon with more than 3 sides will have n(n-3)/2 diagonals.

For instance, let us look at a square. A square has 4 sides and 2 diagonals. Let us apply this formula with n = 4.

We get 4(4-3)/2 = 2 diagonals.

Here is a question on finding the number of diagonals.

If a n-sided convex polygon has 14 diagonals, how many sides does the polygon have?

Any n-sided convex polygon has n(n-3)/2 diagonals.
This polygon has 14 diagonals.
i.e., n(n-3)/2 = 14
Or n(n-3) = 28
Solving for n, we get n = 7.

So, the given polygon has 7 sides.

You can access sample practice questions on Geometry for your GMAT Prep by clicking here

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GMAT PS : Quadratic Equation : Parabola cutting x-axis

Here is a question testing concepts of Quadratic equation and nature of roots of quadratic equation.

If y = x² + dx + 9 does not cut the x-axis, then which of the following could be a possible value of d?
I. 0
II. -3
III. 9

A. III only
B. II only
C. I and II only
D. II and III only
E. I and III only

Correct Answer : Choice C. Values that 'd' could take are 0 or -3

Explanation
The question states that the curve (parabola) does not cut the x-axis.

If y = x² + dx + 9 cuts the x-axis then, the points at which it cuts the x-axis will be the roots of the quadratic equation x² + dx + 9 = 0.

As any point on the x-axis will be a value on the number line, the roots will be real numbers.

However, if the curve does not cut the x-axis, then roots of the quadratic equation will be imaginary.

For a quadratic equation "ax² + bx + c = 0 to have imaginary roots, the discriminant b² - 4*a*c < 0 (the discriminant should be negative).

In this equation, d² - 36 < 0
Or d² < 36
i.e., -6 < d < 6.

Amongst the values given, d = 0 and d = -3 lie in this range.
Hence, choice C.

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.