GMAT Math : DSQ: Understanding the properties of integers

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #11 : Arithmetic

How many negative numbers are in the set   ? 

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.

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.

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:

Statement 1 tells you that  and  are of unlike sign, that  and  are of unlike sign, and that  and  are of unlike sign; that is, exactly one number from each pair is negative. Therefore, there are three negative numbers.

Statement 2 tells you that of , and , there can be either exactly zero or two negative numbers; and that of , and , there can be exactly one or three negative numbers. This means that the number of negative numbers among the six can be as few as one or as many as five.

Example Question #11 : Arithmetic

This six-digit number has two digits missing:

______

If the blanks are to be filled with the same digit, what is that digit?

Statement 1: The number is divisible by 4.

Statement 2: The number is divisible by 3.

Possible Answers:

EITHER 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.

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

If the number is divisible by 4, then the last two digits make a number that is divisible by 4; this allows that digit to be 0, 4, or 8.

If the number is divisible by 3, then its digit sum is divisible by 3. This allows the comon digit to be 1, 4, or 7:

Neither statement alone narrows the common digit to one possibility, but if both statements are true, the only possibility becomes 4.

Example Question #12 : Dsq: Understanding The Properties Of Integers

This five-digit number has two digits missing:

______

If both blanks are two be filled with the same digit, what is that digit?

Statement 1: The number is divisible by 5.

Statement 2: The number is not divisible by 3.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

EITHER 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 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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

There are two different choices we can make for the common digit that would make both statements true, 0 and 5. This makes the last digit 0 or 5, making Statement 1 true, Also, this makes the digit sum either

or 

Either way, the digit sum is not a multiple of 3, and the number itself is not a multiple of 3.

Therefore, the two statements together do not provide enought information to answer the question definitively.

 

Example Question #14 : Properties Of Integers

You are given four numbers, .  You know that exactly one of these numbers does not have value 0. Which number is it?

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.

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.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Neither of these statements help you, individually or together. If three of the four numbers are equal to zero, then both products will equal zero.  There is no way of knowing which of the numbers is non-zero.

Example Question #15 : Properties Of Integers

Let  be positive integers. When is   an odd number?

Possible Answers:

 is never odd.

 is odd if and only if exactly one of  is odd.

 is odd if and only if is odd.

 is always odd.

 is odd if and only if both of  are odd.

Correct answer:

 is odd if and only if is odd.

Explanation:

The square of an integer assumes the same parity (even or odd) as the integer itself, so  is odd if and only if  is odd.  is always even, having even factor 4. The sum of any integer and an even integer has the same parity as the first integer. Therefore,  assumes the same parity as , and, subsequently, the same parity as . In other words,  is odd if and only if  is odd.

Example Question #11 : Properties Of Integers

You are given four numbers, .  How many of these numbers have value 0?

Statement 1: 

Statement 2: 

Possible Answers:

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.

EITHER statement ALONE is 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:

The product of two or more numbers is 0 if and only if at least one of the factors is 0.

If you know the first statement, you know that neither  nor  is equal to 0, and that either , or both are equal to 0. You have narrowed the answer down to one or two.

If you know the second statement, you know that  and  are nonzero, but you know nothing about . You have narrowed the answer down to none or one.

If you know both statements, though, you know that  is the only one of the four numbers equal to zero, so you have answered the question.

Example Question #11 : Dsq: Understanding The Properties Of Integers

What is the last digit in the base eight representation of a number?

Statement 1: The number divided by 8 yields a remainder of 5.

Statement 2: The number divided by 16 yields a remainder of 13.

Possible Answers:

EITHER statement ALONE is 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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

To find the last digit in the base  representation of a number, divide the number by ; the remainder is that digit. If we know the number divided by 8 has remainder 5, we know its last base-eight digit is 5. If we know the number divided by 16 is 13, then we still know that the digit is 5, since the reminder when divided by 8 would be .

Example Question #11 : Arithmetic

Let A-B = C. If B is an integer not equal to 0, is C an integer? 

1. A/B is an integer

2. A*B is an integer 

Possible Answers:

Statements 1 and 2 together are not sufficient.

Each statement alone is sufficient.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Correct answer:

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Explanation:

Let's first look at what the question is asking for. They want us to determine if C is an integer. Since we know that B is an integer, C will be an integer only if A is an integer. If A not an integer, C will not be an integer. So from statements 1 and 2, we want to see if we can prove if A is definitely an integer. 

First, let's try statement 2, which says that A*B is an integer. Let's see what happens if B is 2. If B is 2, then 2.5*2 = 5 is an integer, but 2*2 = 4 is also an integer. So A in this case could be an integer or a not. So statement 2 alone is not sufficient to get our answer. 

Now, let's go back to statement 1, which says A/B is an integer. Let's name another variable x. Let x = A/B. If x=A/B we can rewrite this as A = x*B. But since we know that x is an integer (given in the statement) and B is a non-zero integer (given in the question), A is therefore an integer! (The product of two integers is ALWAYS an integer.) This statement alone is enough to prove that A is an integer. 

Since, as discussed before, A is definitely an integer, and B is an integer. An integer minus an integer will always be an integer. Therefore C is an integer, and statement 1 is sufficient to answer the question. 

Example Question #11 : Dsq: Understanding The Properties Of Integers

 and  are nonzero integers. Is  positive, negative, or zero?

Statement 1:  is negative.

Statement 2:  is even.

Possible Answers:

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

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.

Correct answer:

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

Explanation:

Statement 1 alone provides insufficient information to answer the question; for example, , which is positive, but , which is negative.

Statement 2, however, provides proof that  is positive, since any positive or negative number taken to an even power is positive.

Example Question #20 : Properties Of Integers

Let  and  be positive integers. Is  an integer?

Statement 1: 

Statement 2:  and 

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. 

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

If we know Statement 1, we can rewrite  as follows:

If we know Statement 2, we have a system of linear equations that we can solve to get  and :

 

 or 

 

 

By susbstituting: .

 

Either statement is sufficient to prove  an integer.

 

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