Data Sufficiency Alıştırma Soruları — GMAT — Sayfa 3

Soru 41Soru →

If $x$ and $y$ are positive integers, is the product $xy$ divisible by 4?

(1) $x$ is divisible by 4.
(2) $x + y$ is an odd integer.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

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Cevap: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

The correct answer is that Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (1) establishes that

x

is a multiple of 4, so any integer product

xy

must also be a multiple of 4. Statement (2) only guarantees that one of

x

or

y

is even, which is not enough to determine if the product has two factors of 2.

Adım Adım Çözüm

1

Analyze the question stem target

We need to determine whether

xy

is divisible by 4, given that

x

and

y

are positive integers.

Establishing integer constraints ensures we evaluate divisibility conditions properly.

2

Evaluate Statement (1) independently

If

x

is divisible by 4, then

x = 4k

for some positive integer

k

. Then

xy = (4k)y = 4(ky)

, which is divisible by 4. This gives a definitive YES.

Multiplying any multiple of 4 by an integer results in a multiple of 4. Therefore, Statement (1) alone is sufficient.

3

Evaluate Statement (2) independently

If

x + y

is odd, one integer is even and the other is odd. Case 1: If

x = 4

and

y = 1

,

x+y=5

(odd) and

xy = 4

(divisible by 4 -> YES). Case 2: If

x = 2

and

y = 3

,

x+y=5

(odd) and

xy = 6

(not divisible by 4 -> NO).

Since Statement (2) yields both YES and NO outcomes, it alone is not sufficient.

Anahtar Kavram

Divisibility and Parity Rules in Data Sufficiency

Soru 42Soru →

What is the value of the real number $k$ ?

(1) $4k - 8 = 12$
(2) $k^2 = 25$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Cevabı ve açıklamayı göster

Cevap: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Solving Statement (1) gives

4k = 20

, which simplifies to

k = 5

. Because this is a single, unique value, Statement (1) alone is sufficient. Statement (2) gives

k^2 = 25

, which has two valid solutions (

k = 5

or

k = -5

). Without additional constraints, Statement (2) alone is not sufficient to determine a single value for

k

. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Adım Adım Çözüm

1

Evaluate Statement (1) independently

4k - 8 = 12 \implies 4k = 20 \implies k = 5

Solving the linear equation gives a single, unique value for

k

. Thus, Statement (1) alone is sufficient.

2

Evaluate Statement (2) independently

k^2 = 25 \implies k = 5

or

k = -5

Taking the square root yields two distinct possible values for

k

. Because

k

cannot be uniquely determined, Statement (2) alone is not sufficient.

3

Select the Data Sufficiency decision key

Statement (1) is sufficient, but Statement (2) is not.

This corresponds to the standard option where Statement (1) alone is sufficient.

Anahtar Kavram

Linear vs. Quadratic Degree Sufficiency in Data Sufficiency

Soru 43Soru →

In Data Sufficiency analysis, for any non-zero real numbers $x$ and $y$ with $|x| \neq |y|$ , the target question 'Is $\frac{x^2 - y^2}{xy} > 0$ ?' is algebraically equivalent to the rephrased target question 'Do $x$ and $y$ have opposite signs if and only if $|x| < |y|$ ?'

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

The statement is TRUE.

The quotient

\frac{x^2 - y^2}{xy}

is positive in two distinct cases: when

x, y

have the same sign and

|x| > |y|

(in which case both 'opposite signs' and '

|x| < |y|

' are false, making their biconditional true), or when

x, y

have opposite signs and

|x| < |y|

(in which case both components are true, making their biconditional true). In cases where the quotient is negative, the biconditional evaluates to false. Thus, the rephrased target question is logically and algebraically identical to the original target question.

Adım Adım Çözüm

1

Analyze the sign conditions required for the fraction to be positive.

\frac{x^2 - y^2}{xy} > 0

requires numerator

x^2 - y^2

and denominator

xy

to have the exact same algebraic sign.

A rational expression

\frac{A}{B}

is strictly positive if and only if

A > 0, B > 0

or

A < 0, B < 0

.

2

Evaluate Case 1: Both numerator and denominator are positive (

xy > 0

and

x^2 - y^2 > 0

).

xy > 0 \implies x, y

have SAME signs (Opposite signs = False).

x^2 - y^2 > 0 \implies |x| > |y|

(

|x| < |y|

= False).

The rephrased question tests 'Opposite signs

\iff |x| < |y|

'. Here, False

\iff

False evaluates to TRUE.

3

Evaluate Case 2: Both numerator and denominator are negative (

xy < 0

and

x^2 - y^2 < 0

).

xy < 0 \implies x, y

have OPPOSITE signs (Opposite signs = True).

x^2 - y^2 < 0 \implies |x| < |y|

(

|x| < |y|

= True).

The rephrased question tests 'Opposite signs

\iff |x| < |y|

'. Here, True

\iff

True evaluates to TRUE.

4

Evaluate cases where the fraction is negative to ensure complete equivalence.

If

xy > 0

and

x^2 - y^2 < 0

, we get False

\iff

True (FALSE). If

xy < 0

and

x^2 - y^2 > 0

, we get True

\iff

False (FALSE).

The biconditional statement yields TRUE in exactly the same scenarios where

\frac{x^2 - y^2}{xy} > 0

, establishing strict algebraic equivalence.

Anahtar Kavram

Target Stem Rephrasing via Quotient Sign Decomposition and Logical Biconditional Equivalence

Soru 44Soru →

If $m$ is a real number, is $m$ an even integer?

(1) $2m$ is an even integer.
(2) $m + 1$ is an odd integer.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Cevabı ve açıklamayı göster

Cevap: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Cevap

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statement (1) allows

m

to be any integer because multiplying any integer by

2

results in an even integer. Therefore,

m

could be

3

(odd) or

4

(even), making Statement (1) insufficient. Statement (2) specifies that

m + 1

is an odd integer. Subtracting

1

from an odd integer always yields an even integer, so

m

must be an even integer. This provides a single, definitive 'Yes' answer, making Statement (2) alone sufficient.

Adım Adım Çözüm

1

Analyze the target question stem.

We need to determine whether

m

is an even integer, given that

m

is a real number.

Rephrasing the question clarifies that a definitive 'Yes' or definitive 'No' answer is required.

2

Evaluate Statement (1):

2m

is an even integer.

If

m = 4

, then

2m = 8

(even integer), and

m

is an even integer (Yes). If

m = 3

, then

2m = 6

(even integer), but

m

is an odd integer (No).

Since Statement (1) yields both 'Yes' and 'No' answers, Statement (1) is NOT sufficient.

3

Evaluate Statement (2):

m + 1

is an odd integer.

If

m + 1

is an odd integer, then

m = (m + 1) - 1

. An odd integer minus

1

is always an even integer.

This guarantees that

m

is an even integer, providing a definitive 'Yes'. Thus, Statement (2) ALONE is sufficient.

Anahtar Kavram

Parity and Integer Properties in Data Sufficiency

Soru 45Soru →

In a GMAT Data Sufficiency 'Yes/No' question asking whether a real variable $x > 0$ , a statement that establishes definitively that $x < 0$ is considered INSUFFICIENT because it yields a negative answer to the question stem.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: False

Cevap

False. In a GMAT 'Yes/No' Data Sufficiency question, any statement that yields a definitive 'No' answer is sufficient.

The statement is false because GMAT Data Sufficiency decision logic defines sufficiency as achieving an unambiguous binary answer. Obtaining a conclusive 'No' settles the question entirely, rendering the statement sufficient rather than insufficient.

Adım Adım Çözüm

1

Identify the logical structure of the question stem.

The target question is a 'Yes/No' Data Sufficiency question asking whether

x > 0

.

Understanding whether a question seeks a specific value or a binary condition determines the sufficiency rule.

2

Apply the standard GMAT Data Sufficiency rule for 'Yes/No' questions.

A statement is sufficient if and only if it leads to a consistent 'Yes' in all permissible cases or a consistent 'No' in all permissible cases.

Sufficiency means certainty of outcome, not necessarily an affirmative outcome.

3

Evaluate the outcome of a statement proving

x < 0

.

Since

x < 0

guarantees that

x

is never greater than 0, the answer to 'Is

x > 0

?' is an absolute 'No'.

A guaranteed 'No' provides complete information and solves the question, making the statement sufficient.

Anahtar Kavram

Yes/No Data Sufficiency Decision Rule (Definitive No is Sufficient)

Soru 46Soru →

If $p$ and $q$ are real numbers such that $p \neq 0$ , is $p^2 q < 0$ ?

(1) $|q - 5| = q - 5$ and $q \neq 5$
(2) $p + q = 2$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Cevabı ve açıklamayı göster

Cevap: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Rephrasing the question stem reveals that because

p \neq 0

,

p^2

is strictly positive; thus

p^2 q < 0

is true if and only if

q < 0

. Statement (1) dictates that

|q - 5| = q - 5

with

q \neq 5

, which implies

q > 5

. Because

q

is strictly positive, the answer to 'Is

q < 0

?' is a definitive 'No'. In GMAT Data Sufficiency Yes/No questions, a statement that consistently yields a 'No' answer is sufficient. Statement (2) leaves

q

undetermined (

q

can be positive or negative depending on

p

), so it is insufficient on its own.

Adım Adım Çözüm

1

Rephrase the question stem using algebraic properties.

Since

p

is a non-zero real number,

p^2 > 0

for all valid

p

. Dividing the inequality

p^2 q < 0

by

p^2

yields

q < 0

. The question simplifies to: 'Is

q < 0

?'

Simplifying the target expression reduces cognitive complexity before evaluating the statements.

2

Evaluate Statement (1) independently.

The identity

|k| = k

holds if and only if

k \ge 0

. Thus,

|q - 5| = q - 5 \implies q - 5 \ge 0 \implies q \ge 5

. Combined with

q \neq 5

, we obtain

q > 5

. If

q > 5

, then

q

is strictly positive, meaning

q < 0

is definitively FALSE ('No'). A definitive 'No' answer is sufficient.

In Yes/No Data Sufficiency, any statement that yields a consistent 'Yes' OR a consistent 'No' is sufficient.

3

Evaluate Statement (2) independently.

Given

p + q = 2

, if

p = 1

, then

q = 1 > 0

('No'). If

p = 5

, then

q = -3 < 0

('Yes'). Statement (2) allows both 'Yes' and 'No' outcomes.

Since Statement (2) cannot yield a single definitive Yes/No answer, it is not sufficient.

Anahtar Kavram

Value vs. Yes/No Data Sufficiency Decision Logic

Soru 47Soru →

In GMAT Data Sufficiency logic, if a statement determines a single unique numerical value for a variable $x$ , that statement is sufficient to answer a 'Value' question asking for the exact value of $x$ , but is insufficient to answer a 'Yes/No' question asking whether $x > 5$ .

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: False

Cevap

The statement is False. A single unique value for

x

provides complete information to answer both a 'Value' question for

x

and any 'Yes/No' question regarding

x

.

The correct response is False because a statement that yields a single unique numerical value for a variable

x

provides complete information about

x

. Substituting that single value into the inequality

x > 5

results in either a definitive 'Yes' or a definitive 'No'. Because both definitive outcomes satisfy GMAT Data Sufficiency criteria, the statement is sufficient for both question types.

Adım Adım Çözüm

1

Analyze the condition for sufficiency in a 'Value' Data Sufficiency question.

A 'Value' question requires determining one single, unique value for the variable. Finding a unique value for

x

satisfies this requirement.

If more than one value is possible, the statement is insufficient.

2

Analyze the condition for sufficiency in a 'Yes/No' Data Sufficiency question.

A 'Yes/No' question requires a definitive 'Yes' or a definitive 'No'.

Sufficiency means eliminating all ambiguity in the binary decision.

3

Evaluate the effect of knowing a single unique value of

x

on the inequality

x > 5

.

If

x

is fixed at a single value (e.g.,

x = 7

), the answer is a definitive 'Yes'. If

x

is fixed at another single value (e.g.,

x = 3

), the answer is a definitive 'No'.

In either case, a single value produces a definitive answer with no remaining ambiguity.

Anahtar Kavram

Value vs. Yes/No Data Sufficiency Decision Logic

Soru 48Soru →

If $n$ is a real number, is $n$ an integer?

(1) $n^2 - 3n$ is an integer.
(2) $n^2 + 5n$ is an integer.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Cevabı ve açıklamayı göster

Cevap: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Cevap

Both statements together are sufficient to answer definitively that

n

is an integer, but neither statement alone is sufficient.

Neither statement alone is sufficient because quadratic equations of the form

n^2 - 3n = k

and

n^2 + 5n = m

have non-integer real roots for certain integer values of

k

and

m

. When both statements are combined, subtracting the two equations reveals that

8n

is equal to an integer

p

, so

n = \frac{p}{8}

. Substituting

n = \frac{p}{8}

into

n^2 - 3n

requires

\frac{p(p-24)}{64}

to be an integer. Examining the power of 2 in

p(p-24)

proves that

p

must be divisible by 8, which guarantees that

n

is an integer. Thus, the two statements together provide a definitive 'Yes' answer.

Adım Adım Çözüm

1

Evaluate Statement (1) independently.

Statement (1) is NOT sufficient.

If

n^2 - 3n = -1

(an integer), then

n^2 - 3n + 1 = 0

, giving

n = \frac{3 + \sqrt{5}}{2}

, which is not an integer. However, if

n = 3

,

n^2 - 3n = 0

, which is an integer. Thus,

n

can be an integer or a non-integer.

2

Evaluate Statement (2) independently.

Statement (2) is NOT sufficient.

If

n^2 + 5n = 1

(an integer), then

n^2 + 5n - 1 = 0

, giving

n = \frac{-5 + \sqrt{29}}{2}

, which is not an integer. However, if

n = 1

,

n^2 + 5n = 6

, which is an integer. Thus,

n

can be an integer or a non-integer.

3

Combine Statement (1) and Statement (2).

Deduce that

8n

must be an integer.

Let

n^2 - 3n = k

and

n^2 + 5n = m

, where

k

and

m

are integers. Subtracting the first equation from the second yields

(n^2 + 5n) - (n^2 - 3n) = m - k \implies 8n = m - k

. Since

m - k

is an integer, let

8n = p

for some integer

p

. Hence,

n = \frac{p}{8}

.

4

Analyze whether

n = \frac{p}{8}

can be a non-integer fraction.

n

MUST be an integer, making BOTH statements TOGETHER sufficient.

Substitute

n = \frac{p}{8}

back into Statement (1):

n^2 - 3n = \frac{p^2}{64} - \frac{3p}{8} = \frac{p^2 - 24p}{64}

. For this value to be an integer,

p^2 - 24p = p(p - 24)

must be divisible by

64 = 2^6

. If

p

is odd,

p - 24

is odd, so

p(p-24)

is odd and not divisible by 64. If

p

is a multiple of 2 but not 4 (

p = 2b

with

b

odd),

p(p-24) = 4b(b-12)

, containing only

2^2

factors of 2. If

p

is a multiple of 4 but not 8 (

p = 4b

with

b

odd),

p(p-24) = 16b(b-6)

, containing only

2^4

factors of 2. Thus, for

p(p-24)

to contain

2^6

,

p

must be a multiple of 8. If

p

is a multiple of 8, then

n = \frac{p}{8}

must be an integer.

Anahtar Kavram

Evaluating integer constraints vs. real numbers in Data Sufficiency through algebraic elimination and prime factorization of rational expressions.

Soru 49Soru →

For all non-zero real numbers $x$ and $y$ such that $x \neq y$ , the Data Sufficiency Yes/No target question "Is $\frac{x}{y} + \frac{y}{x} > 2$ ?" is algebraically equivalent to the simplified target question "Is $xy > 0$ ?"

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

True

The statement is True because rewriting the inequality as

\frac{(x - y)^2}{xy} > 0

shows that since

(x - y)^2 > 0

for all

x \neq y

, the inequality holds if and only if

xy > 0

.

Adım Adım Çözüm

1

Combine the fractions on the left-hand side of the target inequality

\frac{x^2 + y^2}{xy} > 2

Finding a common denominator

xy

combines the two reciprocal ratios into a single rational expression.

2

Move all terms to one side to compare against zero

\frac{x^2 + y^2}{xy} - 2 > 0 \implies \frac{x^2 - 2xy + y^2}{xy} > 0

Comparing rational expressions directly to zero simplifies testing for positive or negative regions.

3

Factor the numerator into a perfect square

\frac{(x - y)^2}{xy} > 0

Recognizing

x^2 - 2xy + y^2

as

(x - y)^2

isolates a term with a guaranteed non-negative property.

4

Evaluate sign conditions under the problem constraints

Because

x \neq y

,

(x - y)^2

is strictly positive. Thus,

\frac{(x - y)^2}{xy} > 0

holds if and only if

xy > 0

.

A quotient with a strictly positive numerator is positive if and only if its denominator is positive.

Anahtar Kavram

Question Stem Simplification via Rational Expression Factoring

Soru 50Soru →

If $x$ is a real number, is $x$ an integer?

(1) $x^2 + x$ is an integer.
(2) $x^3 + x^2$ is an integer.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Cevabı ve açıklamayı göster

Cevap: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Cevap

Both statements together are sufficient to determine that

x

is an integer, but neither statement alone is sufficient.

Evaluating each statement independently reveals that non-integer real numbers can produce integer values for

x^2 + x

or

x^3 + x^2

. However, combining both statements allows us to factor

x^3 + x^2

as

x(x^2 + x)

. Setting

x^2 + x = m

and

x^3 + x^2 = k

for integers

m

and

k

, we find

x \cdot m = k

. If

m = 0

,

x

is 0 or -1 (integers). If

m \neq 0

,

x = \frac{k}{m}

is a rational number. Writing

x = \frac{a}{b}

in lowest terms and substituting it back into

x^2 + x = m

shows that

b

must divide

a^2

, forcing

b = 1

. Thus,

x

must be an integer, providing a definitive affirmative answer.

Adım Adım Çözüm

1

Evaluate Statement (1) independently.

Statement (1) is INSUFFICIENT.

If

x = 2

, then

x^2 + x = 6

, which is an integer (YES). However, if

x = \frac{-1 + \sqrt{5}}{2}

, then

x^2 + x - 1 = 0 \implies x^2 + x = 1

, which is an integer, but

x

is not an integer (NO). Thus, Statement (1) alone is insufficient.

2

Evaluate Statement (2) independently.

Statement (2) is INSUFFICIENT.

If

x = 2

, then

x^3 + x^2 = 12

, an integer (YES). If

x

is the real root of

x^3 + x^2 = 1

(where

x \approx 0.755

),

x^3 + x^2

is an integer, but

x

is not an integer (NO). Thus, Statement (2) alone is insufficient.

3

Evaluate Statements (1) and (2) together.

Statements (1) and (2) together are SUFFICIENT.

Let

x^2 + x = m

where

m \in \mathbb{Z}

, and

x^3 + x^2 = k

where

k \in \mathbb{Z}

. Notice that

x^3 + x^2 = x(x^2 + x) = x \cdot m = k

. If

m = 0

, then

x^2 + x = 0 \implies x = 0

or

x = -1

, both of which are integers. If

m \neq 0

, then

x = \frac{k}{m}

, meaning

x

must be a rational number. Express

x = \frac{a}{b}

in lowest terms, where

a, b \in \mathbb{Z}

,

b > 0

, and

\gcd(a, b) = 1

. Substituting

x = \frac{a}{b}

into

x^2 + x = m

yields

\frac{a^2 + ab}{b^2} = m \implies a^2 + ab = m b^2 \implies a^2 = b(mb - a)

. This implies that

b

must divide

a^2

. Since

\gcd(a, b) = 1

,

b

can only divide

a^2

if

b = 1

. Therefore,

x = \frac{a}{1} = a

, which means

x

MUST be an integer. The answer to the question is a definitive YES.

Anahtar Kavram

Testing real vs. integer constraints and applying rational root divisibility properties in Data Sufficiency.

Tahmini Süre:2m 0s

Soru 51Soru →

If $x$ and $y$ are positive real numbers, is $x$ an integer?

(1) $x + y$ is an integer, and $x^2 + xy$ is an integer.
(2) $x - y$ is an integer, and $x^2 - y^2$ is an integer.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Cevabı ve açıklamayı göster

Cevap: Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

Statements (1) and (2) together are not sufficient to determine whether

x

is an integer.

The correct option identifies that statements (1) and (2) together are not sufficient. Testing half-integers such as

x = 2.5

and

y = 1.5

shows that all given conditions (

x+y=4

,

x^2+xy=10

,

x-y=1

,

x^2-y^2=4

) are integers while

x

is not an integer. Conversely,

x = 3

and

y = 1

satisfies all conditions with an integer

x

. Thus, a definitive Yes/No answer cannot be determined.

Adım Adım Çözüm

1

Analyze Statement (1) independently.

Statement (1) states that

x + y = k

(where

k

is an integer) and

x^2 + xy = x(x+y) = m

(where

m

is an integer). Thus,

x \cdot k = m \implies x = \frac{m}{k}

. Since

m

and

k

are integers,

x

is rational, but not necessarily an integer. For example, if

x = 1.5

and

y = 0.5

, then

x + y = 2

(integer) and

x^2 + xy = 1.5(2) = 3

(integer), but

x = 1.5

is not an integer. If

x = 2

and

y = 1

, then

x + y = 3

and

x^2 + xy = 6

, and

x = 2

is an integer. Thus, Statement (1) alone is INSUFFICIENT.

Variable

x

is only constrained to be a rational number

\frac{m}{k}

, which may or may not be an integer.

2

Analyze Statement (2) independently.

Statement (2) states that

x - y = d

(where

d

is an integer) and

x^2 - y^2 = (x-y)(x+y) = p

(where

p

is an integer). Thus

d(x+y) = p \implies x+y = \frac{p}{d}

. Then

2x = (x+y) + (x-y) = \frac{p}{d} + d = \frac{p + d^2}{d} \implies x = \frac{p + d^2}{2d}

. If

x = 2.5

and

y = 1.5

, then

x - y = 1

(integer) and

x^2 - y^2 = 6.25 - 2.25 = 4

(integer), but

x = 2.5

is not an integer. If

x = 2

and

y = 1

,

x - y = 1

and

x^2 - y^2 = 3

, and

x = 2

is an integer. Thus, Statement (2) alone is INSUFFICIENT.

Variable

x

is constrained to a rational form with denominator

2d

, which allows non-integer values.

3

Analyze Statements (1) and (2) combined.

Combining both statements gives

x + y = k

(integer) and

x - y = d

(integer). Adding these yields

2x = k + d

, so

2x

must be an integer. This means

x

can be an integer or a half-integer (e.g.,

2.5

). Testing

x = 2.5

and

y = 1.5

: both are positive real numbers;

x + y = 4

(integer);

x^2 + xy = 2.5(4) = 10

(integer);

x - y = 1

(integer); and

x^2 - y^2 = 6.25 - 2.25 = 4

(integer). Here

x

is NOT an integer. Testing

x = 3

and

y = 1

: both are positive real numbers;

x + y = 4

(integer);

x^2 + xy = 3(4) = 12

(integer);

x - y = 2

(integer); and

x^2 - y^2 = 9 - 1 = 8

(integer). Here

x

IS an integer. Because both 'Yes' and 'No' answers are possible under all conditions, Statements (1) and (2) together are INSUFFICIENT.

The system of equations allows

x

to be a half-integer (such as

2.5

), satisfying all integer conditions of both statements without requiring

x

itself to be an integer.

Anahtar Kavram

Evaluating implicit non-integer constraints and system degree in Data Sufficiency questions.

Soru 52Soru →

If $k$ is a positive real number, is $k$ an integer?

(1) $\sqrt{k}$ is an integer.
(2) $k^2 - 4k$ is an integer.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Cevabı ve açıklamayı göster

Cevap: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (1) alone is sufficient because if the square root of

k

is an integer

m

, then

k = m^2

, which is guaranteed to be an integer. Statement (2) alone is not sufficient because

k^2 - 4k

can equal an integer when

k = 4

(an integer) or when

k = 2 + \sqrt{5}

(a non-integer). Therefore, only Statement (1) is sufficient.

Adım Adım Çözüm

1

Evaluate Statement (1) independently.

If

\sqrt{k} = m

where

m

is an integer, then

k = m^2

. Since

m

is an integer,

m^2

must also be an integer. Thus,

k

is definitely an integer (Yes).

The square of any integer is always an integer, so Statement (1) is sufficient.

2

Evaluate Statement (2) independently.

Let

k^2 - 4k = n

, where

n

is an integer. If

n = 0

, then

k(k - 4) = 0 \implies k = 4

(an integer). If

n = 1

, then

k^2 - 4k - 1 = 0 \implies k = 2 + \sqrt{5}

(not an integer).

Since

k

can be either an integer or a non-integer, Statement (2) does not provide a definitive Yes/No answer and is not sufficient.

Anahtar Kavram

Algebraic definitions of integers versus real numbers in Data Sufficiency

Tahmini Süre:2m 0s

Soru 53Soru →

If $k$ and $m$ are non-zero real numbers, is $k > m$ ?

(1) $k^2 > m^2$
(2) $\frac{k - m}{m^2 + 1} > 0$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Cevabı ve açıklamayı göster

Cevap: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Cevap

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

The target question asks if

k > m

. Evaluating Statement (1) alone:

k^2 > m^2

indicates

|k| > |m|

, which permits cases where

k > m

(e.g.,

k=3, m=2

) and cases where

k < m

(e.g.,

k=-5, m=2

). Thus, Statement (1) is insufficient. Evaluating Statement (2) independently:

\frac{k - m}{m^2 + 1} > 0

. For any real number

m

,

m^2 + 1

is strictly positive. Multiplying both sides of the inequality by

(m^2 + 1)

preserves the inequality sign, yielding

k - m > 0

, which simplifies directly to

k > m

. This yields a definitive 'Yes' answer. Therefore, Statement (2) alone is sufficient.

Adım Adım Çözüm

1

Rephrase the target question

The target question asks whether the difference

(k - m)

is strictly positive.

Determining if

k > m

is algebraically equivalent to confirming

k - m > 0

.

2

Evaluate Statement (1) independently

Statement (1) states

k^2 > m^2

, which implies

|k| > |m|

. If

k = 3

and

m = 2

, then

9 > 4

and

3 > 2

(Yes). However, if

k = -5

and

m = 2

, then

25 > 4

but

-5 < 2

(No).

Because both 'Yes' and 'No' outcomes are possible, Statement (1) alone is not sufficient.

3

Evaluate Statement (2) independently without carrying over Statement (1)

Statement (2) states

\frac{k - m}{m^2 + 1} > 0

. Since

m

is a real number,

m^2 \ge 0

, which guarantees

m^2 + 1 \ge 1 > 0

. Multiplying the inequality by the strictly positive quantity

(m^2 + 1)

preserves the inequality direction, yielding

k - m > 0

, or

k > m

.

This yields a definitive 'Yes' answer, so Statement (2) alone is sufficient.

Anahtar Kavram

Statement Independence in Data Sufficiency

Soru 54Soru →

If $p$ and $q$ are non-zero real numbers such that $p \neq q$ , is $\frac{p^3 q - p q^3}{p - q} > 0$ ?

(1) $pq > 0$
(2) $p + q > 0$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Cevabı ve açıklamayı göster

Cevap: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Cevap

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Factoring the numerator gives

p^3 q - p q^3 = pq(p-q)(p+q)

. Since

p \neq q

, dividing by

(p-q)

yields

pq(p+q)

. To determine whether

pq(p+q) > 0

, we need the sign of the product

pq(p+q)

. Statement (1) tells us

pq > 0

, which is insufficient by itself because

p+q

could be positive or negative. Statement (2) tells us

p+q > 0

, which is insufficient by itself because

pq

could be positive or negative. Together, both terms

pq

and

p+q

are positive, so their product must be positive, providing a definitive 'Yes'.

Adım Adım Çözüm

1

Rephrase the target question stem algebraically by factoring the numerator.

\frac{p^3 q - p q^3}{p - q} = \frac{p q (p^2 - q^2)}{p - q} = \frac{p q (p - q)(p + q)}{p - q} = pq(p + q)

. The target question simplifies to: 'Is

pq(p+q) > 0

?'

Simplifying the stem isolates the specific product whose sign determines sufficiency.

2

Evaluate Statement (1):

pq > 0

.

If

p = 1, q = 1

, then

pq = 1 > 0

and

pq(p+q) = 1(2) = 2 > 0

(Yes). If

p = -1, q = -1

, then

pq = 1 > 0

and

pq(p+q) = 1(-2) = -2 < 0

(No).

Knowing

pq > 0

leaves the sign of

p+q

unknown, so Statement (1) alone is NOT sufficient.

3

Evaluate Statement (2):

p + q > 0

.

If

p = 2, q = 1

, then

p+q = 3 > 0

and

pq(p+q) = 2(3) = 6 > 0

(Yes). If

p = 3, q = -1

, then

p+q = 2 > 0

and

pq(p+q) = (-3)(2) = -6 < 0

(No).

Knowing

p+q > 0

leaves the sign of

pq

unknown, so Statement (2) alone is NOT sufficient.

4

Evaluate Statement (1) and Statement (2) together.

From Statement (1),

pq > 0

. From Statement (2),

p+q > 0

. Therefore,

pq(p+q) > 0

as it is the product of two strictly positive real numbers.

Combining both statements provides a definitive 'Yes' answer to the simplified target question.

Anahtar Kavram

Question Stem Simplification and Target Rephrasing

Soru 55Soru →

If $a$ and $b$ are non-zero real numbers, is $a^3 b + a b^3 > 0$ ?

(1) $a^2 - 5ab + 6b^2 = 0$
(2) $|a + b| < |a - b|$

Which of the following choices correctly describes the sufficiency of the statements?

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Cevabı ve açıklamayı göster

Cevap: EACH statement ALONE is sufficient.

Cevap

EACH statement ALONE is sufficient.

Rephrasing the stem shows that

a^3b + ab^3 = ab(a^2 + b^2) > 0

reduces to asking whether

ab > 0

. Statement (1) yields

a = 2b

or

a = 3b

, both of which make

ab > 0

(a definitive 'Yes'). Statement (2) simplifies to

4ab < 0 \implies ab < 0

, which yields a definitive 'No'. Because each statement independently produces a definitive, consistent Yes/No answer, each statement alone is sufficient.

Adım Adım Çözüm

1

Rephrase the question stem target.

The target expression

a^3b + ab^3

factors into

ab(a^2 + b^2)

. Because

a

and

b

are non-zero real numbers,

a^2 + b^2 > 0

always. Thus,

a^3b + ab^3 > 0

if and only if

ab > 0

. The question reduces to a Yes/No query: 'Is

ab > 0

?'

Simplifying the stem isolates the essential logical condition needed to evaluate sufficiency.

2

Evaluate Statement (1) alone.

Factoring

a^2 - 5ab + 6b^2 = 0

yields

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

, meaning

a = 2b

or

a = 3b

. If

a = 2b

, then

ab = 2b^2 > 0

(since

b \neq 0

). If

a = 3b

, then

ab = 3b^2 > 0

(since

b \neq 0

). In both cases,

ab > 0

is true, giving a definitive 'Yes'.

Although Statement (1) yields two possible values for the ratio

a/b

, both values guarantee that

ab > 0

. In Yes/No Data Sufficiency, a consistent 'Yes' is sufficient.

3

Evaluate Statement (2) alone.

Squaring both non-negative sides of

|a + b| < |a - b|

gives

(a + b)^2 < (a - b)^2

, which simplifies to

a^2 + 2ab + b^2 < a^2 - 2ab + b^2 \implies 4ab < 0 \implies ab < 0

. Since

ab < 0

, the answer to 'Is

ab > 0

?' is a definitive 'No'.

A statement that yields a consistent, definitive 'No' is sufficient in a Yes/No Data Sufficiency question.

4

Synthesize results.

Since Statement (1) alone produces a definitive 'Yes' and Statement (2) alone produces a definitive 'No', each statement alone is sufficient to answer the question.

Both statements independently satisfy the decision criteria for Data Sufficiency.

Anahtar Kavram

Value vs. Yes/No Data Sufficiency Decision Logic

Soru 56Soru →

If $a$ and $b$ are real numbers, is $a^2 - b^2 > 0$ ?

(1) $|a| < b$
(2) $a + b > 0$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Cevabı ve açıklamayı göster

Cevap: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (1) establishes that

|a| < b

, which implies

b > 0

and

a^2 < b^2

. This guarantees that

a^2 - b^2 < 0

, leading to a definitive 'No' answer to the question 'Is

a^2 - b^2 > 0

?'. In GMAT Data Sufficiency Yes/No questions, a statement that consistently leads to a 'No' answer is sufficient. Statement (2) provides

a + b > 0

, which allows for cases where

a^2 - b^2 > 0

(e.g.,

a=3, b=1

) and cases where

a^2 - b^2 < 0

(e.g.,

a=1, b=3

), making it insufficient.

Adım Adım Çözüm

1

Analyze the target question stem

The question asks whether

a^2 - b^2 > 0

, which is equivalent to asking if

|a| > |b|

or if

a^2 > b^2

. This is a Yes/No Data Sufficiency question.

Establishing what condition constitutes a 'Yes' versus a 'No' is essential before evaluating individual statements.

2

Evaluate Statement (1):

|a| < b

Since

|a| \ge 0

, it follows that

b > 0

. Squaring both non-negative sides of

|a| < b

gives

|a|^2 < b^2

, which simplifies to

a^2 < b^2

. Subtracting

b^2

yields

a^2 - b^2 < 0

.

Because

a^2 - b^2 < 0

for all values satisfying Statement (1), the answer to 'Is

a^2 - b^2 > 0

?' is a definitive 'No'. In Data Sufficiency, a definitive 'No' means Statement (1) is ALONE sufficient.

3

Evaluate Statement (2):

a + b > 0

If

a = 3

and

b = 1

, then

a + b = 4 > 0

and

a^2 - b^2 = 9 - 1 = 8 > 0

(Yes). If

a = 1

and

b = 3

, then

a + b = 4 > 0

and

a^2 - b^2 = 1 - 9 = -8 < 0

(No).

Since Statement (2) yields both 'Yes' and 'No' responses depending on the specific values chosen, Statement (2) is NOT sufficient.

4

Determine the final Data Sufficiency decision logic

Statement (1) alone is sufficient to yield a definitive 'No' answer, while Statement (2) alone is not sufficient.

In GMAT Data Sufficiency Yes/No questions, any statement that provides a single, conclusive answer (whether 'Yes' or 'No') is sufficient.

Anahtar Kavram

Definitive No Sufficiency in Yes/No Data Sufficiency

Tahmini Süre:2m 0s

Soru 57Soru →

If $x$ is a real number, is $|x - 3| < 5$ ?

(1) $1 < x < 6$
(2) $x > 0$

Which of the following describes the sufficiency of the statements?

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Cevabı ve açıklamayı göster

Cevap: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

The correct choice is the option stating that Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Rephrasing

|x - 3| < 5

yields

-2 < x < 8

. Statement (1) restricts

x

to

(1, 6)

, which is completely inside

(-2, 8)

, guaranteeing a 'Yes' answer. Statement (2) allows values like

x = 2

('Yes') and

x = 10

('No'), so it is not sufficient.

Adım Adım Çözüm

1

Rephrase the target inequality from the stem.

The absolute value inequality

|x - 3| < 5

is equivalent to

-5 < x - 3 < 5

. Adding 3 to all parts gives

-2 < x < 8

. The question asks: 'Is

x

strictly between

-2

and

8

?'

Simplifying the stem into a clear range for

x

makes evaluating sufficiency straightforward.

2

Evaluate Statement (1):

1 < x < 6

.

Every number in the interval

(1, 6)

is strictly between

-2

and

8

. Thus, the answer to 'Is

-2 < x < 8

?' is a definitive 'Yes'. Statement (1) alone is sufficient.

Since the entire set of allowed values under statement (1) is a subset of

(-2, 8)

, statement (1) provides a conclusive answer.

3

Evaluate Statement (2):

x > 0

.

If

x = 2

, then

-2 < 2 < 8

is True (Yes). However, if

x = 10

, then

-2 < 10 < 8

is False (No). Because both 'Yes' and 'No' are possible, Statement (2) alone is not sufficient.

A statement is insufficient if it permits values that produce conflicting answers to the target question.

Anahtar Kavram

Absolute Value Inequalities and Range Containment in Data Sufficiency

Soru 58Soru →

In a GMAT Data Sufficiency 'Yes/No' question, if a statement permits multiple numerical values for a variable, but every permitted value yields a conclusive 'No' to the question stem, the statement is sufficient to answer the question.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

True

In GMAT Data Sufficiency 'Yes/No' questions, sufficiency requires a single, conclusive answer to the question asked. A statement that guarantees a 'No' answer under all valid cases is completely sufficient, even if the underlying variables can take on multiple numerical values.

Adım Adım Çözüm

1

Analyze the core rule of Yes/No Data Sufficiency questions.

A statement is sufficient if it leads to a definitive, unambiguous 'Yes' or a definitive, unambiguous 'No' answer.

Data Sufficiency evaluates whether the provided data leads to one consistent answer.

2

Evaluate the condition described in the statement.

Although multiple numerical values are possible, all permitted values consistently yield 'No'.

Because there are no scenario outcomes that yield 'Yes', there is zero ambiguity in answering the question stem.

3

Conclude sufficiency.

The statement is sufficient.

A consistent 'No' satisfies the requirement of data sufficiency just as effectively as a consistent 'Yes'.

Anahtar Kavram

Definitive Yes/No Sufficiency Rule

Soru 59Soru →

If $x$ and $y$ are positive real numbers, is $\frac{1}{x} < \frac{1}{y}$ ?

(1) $x^2 > y^2$
(2) $x - y > 0$

Which of the following describes the sufficiency of the statements to answer the question?

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Cevabı ve açıklamayı göster

Cevap: EACH statement ALONE is sufficient.

Cevap

EACH statement ALONE is sufficient.

Rephrasing the question stem

\frac{1}{x} < \frac{1}{y}

for positive numbers

x

and

y

yields

x > y

. Statement (1) states

x^2 > y^2

, which for positive numbers implies

x > y

(sufficient). Statement (2) states

x - y > 0

, which simplifies directly to

x > y

(sufficient). Therefore, each statement alone is sufficient.

Adım Adım Çözüm

1

Rephrase the question stem target inequality.

Since

x

and

y

are both positive, multiplying

\frac{1}{x} < \frac{1}{y}

by

xy

preserves the inequality sign, yielding

y < x

, which is equivalent to

x > y

. The target question simplifies to: 'Is

x > y

?'

Simplifying the target question stem allows direct evaluation of each statement.

2

Evaluate Statement (1):

x^2 > y^2

.

Since

x

and

y

are positive real numbers, taking the positive square root of both sides of

x^2 > y^2

yields

x > y

. This gives a definitive 'Yes' to the rephrased question.

Statement (1) alone provides sufficient information.

3

Evaluate Statement (2):

x - y > 0

.

Adding

y

to both sides of

x - y > 0

gives

x > y

. This gives a definitive 'Yes' to the rephrased question.

Statement (2) alone provides sufficient information.

Anahtar Kavram

Question Stem Simplification and Target Rephrasing

Soru 60Soru →

If $x$ and $y$ are real numbers, is $x > y$ ?

(1) $y - x = |x + 3| + 1$
(2) $x + y = 10$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Cevabı ve açıklamayı göster

Cevap: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

The option stating that Statement (1) alone is sufficient while Statement (2) alone is not is correct. Statement (1) demonstrates that

y - x = |x + 3| + 1 \ge 1

, which proves

y > x

. Consequently, the answer to 'Is

x > y

?' is a guaranteed, definitive 'No'. In GMAT Yes/No Data Sufficiency questions, any statement that yields a conclusive 'No' is sufficient. Statement (2) allows multiple relative orderings of

x

and

y

, making it insufficient.

Adım Adım Çözüm

1

Analyze the question stem

The target is a Yes/No question: 'Is

x > y

?' Any statement that yields a definitive 'Yes' OR a definitive 'No' is sufficient.

Data Sufficiency Yes/No decision logic requires evaluating whether a statement yields a single consistent truth value.

2

Evaluate Statement (1):

y - x = |x + 3| + 1

Since absolute value is always non-negative (

|x + 3| \ge 0

), we have

y - x \ge 0 + 1 = 1 > 0

. Thus,

y - x > 0 \implies y > x

.

Because

y > x

is always true under Statement (1), the answer to 'Is

x > y

?' is a definitive 'No'.

3

Conclude sufficiency for Statement (1)

A definitive 'No' is SUFFICIENT. Statement (1) alone is sufficient.

In GMAT Data Sufficiency, a consistent negative answer resolves the question completely.

4

Evaluate Statement (2):

x + y = 10

If

x = 6

and

y = 4

, then

x > y

is 'Yes'. If

x = 4

and

y = 6

, then

x > y

is 'No'.

Since both 'Yes' and 'No' are possible, Statement (2) alone is NOT sufficient.

Anahtar Kavram

Definitive Yes/No Decision Logic in Data Sufficiency

Data Sufficiency

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