GMAT Math : DSQ: Understanding real numbers

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #1250 : Data Sufficiency Questions

Evaluate .

Statement 1:  is the multiplicative inverse of .

Statement 2:  is the additive inverse of .

Possible Answers:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Correct answer:

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Explanation:

Assume Statement 1 alone. The product of a number and its multiplicative inverse is 1, so , and . But without knowing anything else, the expression cannot be evaluated.

Assume Statement 2 alone. The sum of a number and its additive inverse is 0, so . By the distributive property,

Example Question #1251 : Data Sufficiency Questions

 is a real number. True or false:  is a rational number.

Statement 1:  is a rational number.

Statement 2:  is a rational number.

Possible Answers:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Explanation:

Statement 1 provides insufficient information to determine whether  is rational or not.

If , which, being an integer, is rational, then , which, being an integer, is rational.

If . which is irrational, then , which, being an integer, is rational.

Statement 2 is, however, sufficient. If  for some rational number , then . 1, being an integer, is rational, and as the difference of rational numbers,  is rational.

Example Question #1252 : Data Sufficiency Questions

 is a real number. True or false:  is a rational number.

Statement 1:  is an irrational number.

Statement 2:  is an irrational number.

Possible Answers:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Assume Statement 1 alone. , the quotient of integers, is rational. The sum of any two rational numbers is rational, so if  is rational, then  is a rational number. However,  is irrational, so, contrapositively,  is irrational. 

Assume Statement 2 alone. , the quotient of integers, is rational. The product of any two rational numbers is rational, so if  is rational, then  is a rational number. However,  is irrational, so, contrapositively,  is irrational. 

Example Question #1253 : Data Sufficiency Questions

Evaluate .

Statement 1:  is the additive inverse of .

Statement 2:  is the additive inverse of .

Possible Answers:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Explanation:

Assume both statements to be true. The additive inverse of a number is the number that can be added to that number to yield sum 0. We show that the value of  cannot be determined from these two statements by examining two cases:

Case 1: 

 and , so  is the additive inverse of  and  is the additive inverse of 

.

Case 2: 

 and , so  is the additive inverse of  and  is the additive inverse of 

In both scenarios, the conditions of both statements are met, but  assumes different values. The two statements together are insufficient to answer the question.

Example Question #21 : Real Numbers

True or false: is an integer.

Statement 1: The multiplicative inverse of is not an integer.

Statement 2: The additive inverse of is an integer.

Possible Answers:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Correct answer:

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Explanation:

Assume Statement 1 alone. The multiplicative inverse of a number is the number which, when multiplied by that number, yields a product of 1. If the multiplicative inverse of is not an integer, it is possible for to be an integer or to not be one, as is shown in these examples:

If , which is an integer, then, since , the multiplicative inverse of is , which is not an integer, If , which is not an integer, then, since , the multiplicative inverse of is , which is not an integer.

Assume Statement 2 alone. The additive inverse of a number is the number which, when added to that number, yields a sum of 0. If is the additive inverse of the number , then

, or

by Statement 2, is an integer; ., the product of integers, is itself an integer.

 

Example Question #1255 : Data Sufficiency Questions

Define an operation  on the real numbers as follows:

If both  and  are whole numbers, then .

If  and  are not both whole numbers, then .

Evaluate .

Statement 1:  is not an integer.

Statement 2: .

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

Assume Statement 1 alone. Every whole number (0, 1, 2, 3,...) is an integer, so if  is not an integer, it cannot be a whole number. Therefore, since  and  are not whole numbers, the second defintion of  is used, and .

Assume Statement 2 alone. , which is a whole number, so it is not clear what definition of  is used. If  is not a whole number, the second defintion of  is used, and . If  is also a whole number, then the first defintion is used:

.

However, we do not know the value of .

Overall, Statement 2 provides insufficient information to answer the question.

Example Question #1256 : Data Sufficiency Questions

Define an operation  on the positive integers as follows:

If  and  are both prime integers, then .

If  and  are not both prime integers, then .

Evaluate .

Statement 1: .

Statement 2:  is a factor of .

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

Assume Statement 1 alone. A prime number is an integer with exaclty two factors - 1 and the number itself. 1 is considered to not be prime, since it has one factor. Therefore, if ,  and are not both prime, and the second defintion of  is used. .

Assume Statement 2 alone. 

Case 1: .

In this case, Statement 1 is true, and as demonstrated before, .

Case 2: .

In this case,   is a factor of , since any integer is a factor of itself. Both integers are prime, so the first defintion of  is used. .

Example Question #1257 : Data Sufficiency Questions

Define an operation  on the real numbers as follows:

If both  and  are both positive, then .

If both  and  are not both positive, then .

Evaluate .

Statement 1: .

Statement 2: .

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

Assume Statement 1 alone. , so  and  are each other's opposites. Either one is positive and one is negative, or both are equal to 0; either way, the definition of  for  and  not both positive is used, and 

.

Assume Statement 2 alone. Again, the definition of  for  and  not both positive is used, and

.

However, we do not know the value of .

Example Question #1258 : Data Sufficiency Questions

Define an operation  on the positive integers as follows:

If  and  are both prime integers, then .

If  and  are not both prime integers, then .

Evaluate .

Statement 1: .

Statement 2:  is a factor of .

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Assume Statement 1 alone. A prime number is an integer with exaclty two factors - 1 and the number itself. 2 has only two factors, 1, and 2, and is therefore prime. However, we do not know . If  is not prime, then . If  is prime, then , which cannot be calculated without knowing .

Assume Statement 2 alone.

Unless both numbers are primes, the second definition of  is used, and .

The only way for both numbers to be primes and  to be a factor of  is for both  and  to be the same prime number. If this is the case, the first definition is used:

.

Therefore,  regardless.

Example Question #1259 : Data Sufficiency Questions

Define an operation  on the real numbers as follows:

If both  and  are integers, then .

If  and  are not both integers, then .

Evaluate .

Statement 1: 

Statement 2:  and 

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. The defintion that will be used to calculate  is not clear, since it is not known whether the numbers are integers. For example, if

,

since both are integers, the definition  will be used, and

.

If , the defintion . will be used, and

.

Assume Statement 2 alone. Both numbers fall between two consecutive integers, so neither is an integer, and the definition  will be used. However, the value of  cannot be calculated, since , and their difference are unknown.

Now assume both statements to be true. From Statement 2, the definition  will be used. Since ,

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