DSQ: Understanding the properties of integers - GMAT Quantitative
Card 1 of 392
True or false: Positive integer 
is prime.
Statement 1: The last digit of 
is 4.
Statement 2: 
.
True or false: Positive integer
Statement 1: The last digit of
Statement 2:
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Assume Statement 1 alone. An integer is divisible by 2 if and only if its last digit is 0, 2, 4, 6, or 8. Therefore, it therefore follows that 
is divisible by 2. Since the only prime number divisible by 2 is 2 itself, any integer ending in 4 must not be prime.
Assume Statement 2 alone. Then:
Either
, in which case 
- a prime number, or
, in which case 
- a composite number since it has more than two factors (1, 2, 7, 14).
Therefore, it is not clear whether or not 
is prime.
Assume Statement 1 alone. An integer is divisible by 2 if and only if its last digit is 0, 2, 4, 6, or 8. Therefore, it therefore follows that
Assume Statement 2 alone. Then:
Either
Therefore, it is not clear whether or not
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Is 
odd?
(1) 
is odd
(2) 
is even
Is
(1)
(2)
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For statement (1), we only know that 
is odd but we have no idea about 
. If 
is odd, then 
is even. If 
is even, then 
is odd. Therefore we have no clear answer to the question using this condition. For statement (2), since 
is even, we know that 
and 
are either both odd or both even, therefore we know for sure that 
is even and the answer to this question is “no”.
For statement (1), we only know that
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If 
is an integer and 
, what is the value of 
?
(1) 
is a factor of 20.
(2) 
is a factor of 24.
If
(1)
(2)
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From statement (1), we know that the possible value of 
would be 4 and 5. From statement (2), we know that the possible value of 
would be 4 and 6. Putting the two statements together, we know that only 
satisfies both conditions. Therefore both statements together are sufficient.
From statement (1), we know that the possible value of
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What is the remainder when the two digit, positive integer 
is divided by 3?
(1) The sum of the digits of 
is 3.
(2) The difference of the digits is 3.
What is the remainder when the two digit, positive integer
(1) The sum of the digits of
(2) The difference of the digits is 3.
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For statement (1), there are 3 possible two digit positive integers: 30, 12, 21. The remainder when these numbers are divided by 3 is 0. Therefore, statement (1) is sufficient. For statement (2), there are lots of different combinations of integers that have different remainders when divided by 3. For example, the remainder of 30 is 0, but the remainder of 14 is 2. Therefore, statement (2) is insufficient.
For statement (1), there are 3 possible two digit positive integers: 30, 12, 21. The remainder when these numbers are divided by 3 is 0. Therefore, statement (1) is sufficient. For statement (2), there are lots of different combinations of integers that have different remainders when divided by 3. For example, the remainder of 30 is 0, but the remainder of 14 is 2. Therefore, statement (2) is insufficient.
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If 
is a positive integer, is 
divisible by 6?
1. The sum of the digits of 
is divisible by 6
2. 
is even
If
1. The sum of the digits of
2.
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Statement 1: Numbers whose digits sum to a number divisible by 3 are divisible by 3, but the same does not apply to sums of 6. This indicates that 
is divisble by 3 but is not sufficient at proving 
is divisible by 6.
Statement 2: Though all multiples of 6 are even, not all even numbers are multiples of 6.
Together: The fact that 
is a multiple of 3 and even is sufficient evidence for the conclusion that x is divisible by 6.
Statement 1: Numbers whose digits sum to a number divisible by 3 are divisible by 3, but the same does not apply to sums of 6. This indicates that
Statement 2: Though all multiples of 6 are even, not all even numbers are multiples of 6.
Together: The fact that
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, 
, and 
are integers. Is 
odd?
(1) 
is a prime number and 
(2) 
is odd
(1)
(2)
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Statement (1) tells us that 
is an odd number. Since we don’t know if 
and 
are odd numbers, we cannot decide the sign of 
.
Statement (2) tells us that 
is an even number, since 
. When we do the multiplication, the product will be even if one or more of the integers are even. So with statement (2), we know for sure that 
is even.
Statement (1) tells us that
Statement (2) tells us that
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Is 
positive, negative, or zero?
-
is positive.
-
is positive.
Is
-
-
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raised to an odd power must have the same sign as 
, so, if 
is positive, then 
is also positive. But either a positive number or a negative number raised to an even power must be positive. Therefore, 
being positive is inconclusive.
Therefore, the correct choice is that Statement 1, but not Statement 2, is sufficient.
Therefore, the correct choice is that Statement 1, but not Statement 2, is sufficient.
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The greatest common factor of 32 and a number 
is 16. What is 
?
-
3 is also a factor of 
.
-
5 is also a factor of 
.
The greatest common factor of 32 and a number
-
3 is also a factor of
-
5 is also a factor of
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That 
cannot be determined, even if both statements are known to be true, can be proved by demonstrating two examples of 
that fit these conditions. We can do this by comparing the prime factorizations of 32 and 
.
Example: 
To find 
:
Example: 
To find 
:
In each case, 3 and 5 are factors of 
, and in each case, 
.
The answer is that both statements together are insufficient.
That
Example:
To find
Example:
To find
In each case, 3 and 5 are factors of
The answer is that both statements together are insufficient.
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What is the area of a rhombus in square inches?
-
One of its angles measures 
-
One of its sides measures 10 inches
What is the area of a rhombus in square inches?
-
One of its angles measures
-
One of its sides measures 10 inches
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As is true of any other parallelogram, the area of the rhombus is the base multiplied by the height. The common sidelength alone can be used to determine the base, but without the angles, the height cannot be determined. Using trigonometry, the angle can be used to determine the height relative to the base, but without the base, the height is unknown.
If we know both of the given statements, then part of one base, an altitude from an endpoint of the opposite base, and one adjacent side form a 30-60-90 triangle. The hypotenuse of that triangle is 10 inches, and the altitude is half that, or 5 inches. This makes the area 50 square inches.
The answer is that both statements together are sufficient to answer the question, but neither statement alone.
As is true of any other parallelogram, the area of the rhombus is the base multiplied by the height. The common sidelength alone can be used to determine the base, but without the angles, the height cannot be determined. Using trigonometry, the angle can be used to determine the height relative to the base, but without the base, the height is unknown.
If we know both of the given statements, then part of one base, an altitude from an endpoint of the opposite base, and one adjacent side form a 30-60-90 triangle. The hypotenuse of that triangle is 10 inches, and the altitude is half that, or 5 inches. This makes the area 50 square inches.
The answer is that both statements together are sufficient to answer the question, but neither statement alone.
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Data Sufficiency Question
Out of 100 students, 60 took French and 25 took German. How many students took neither?
1. 15 students took Spanish
2. 7 students took both French and German
Data Sufficiency Question
Out of 100 students, 60 took French and 25 took German. How many students took neither?
1. 15 students took Spanish
2. 7 students took both French and German
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Statement 1 does not tell us anything about the number of students taking French or German. The information from statement 2 is sufficient, if 60 took French, 25 took German, and 7 took both, we can calculate the number that took neither.
Statement 1 does not tell us anything about the number of students taking French or German. The information from statement 2 is sufficient, if 60 took French, 25 took German, and 7 took both, we can calculate the number that took neither.
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Does the integer 
have at least 4 different positive prime factors?
(1) 
is an integer.
(2) 
is an integer.
Does the integer
(1)
(2)
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(1) This statement tells us that 2 and 7 are prime factors of 
. This information is not sufficient.
(2) This statement tells us that 3 and 5 are prime factors of 
. this information is not sufficient.
Considering both (1) and (2), we have four positive prime factors of 
between them.
(1) This statement tells us that 2 and 7 are prime factors of
(2) This statement tells us that 3 and 5 are prime factors of
Considering both (1) and (2), we have four positive prime factors of
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If 
and 
are both integers, evaluate 
.
Statement 1: 
.
Statement 2: 
and 
are both prime integers.
If
Statement 1:
Statement 2:
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There are infinitely many primes, and several integers between 13 and 23, so knowing just one of these statements is not enough. But only two integers in the stated range - 17 and 19 - are prime, so knowing both statements tells you that 
and 
are 17 and 19, respectively. Subsequently, you can add them to get 36.
There are infinitely many primes, and several integers between 13 and 23, so knowing just one of these statements is not enough. But only two integers in the stated range - 17 and 19 - are prime, so knowing both statements tells you that
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If a positive integer 
is divided by 2, what is the remainder?
Statement 1: If 
is divided by 2, the remainder is 1.
Statement 2: If 
is divided by 4, the remainder is 3.
If a positive integer
Statement 1: If
Statement 2: If
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The question is the same as asking whether 
is odd or even.
Statement 1 says that the square of 
is odd. If we know this, then we know that 
is odd, since the square of an even number is even.
Statement 2 says that 
is 3 greater than a multiple of 4; this makes 
odd.
Therefore, either statement alone tells us that 
is an odd number.
The question is the same as asking whether
Statement 1 says that the square of
Statement 2 says that
Therefore, either statement alone tells us that
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You are given two positive integers, 
and 
. Is their product divisible by 7?
Statement 1: 
is divisible by 7.
Statement 2: 
is divisible by 7.
You are given two positive integers,
Statement 1:
Statement 2:
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If we only know that 
is divisible by 7, then 
may or may not be a multiple of 7.
Example 1: If 
, then 
, which is not divisible by 7.
Example 2: If 
, then 
, which is divisible by 7.
A similar argument shows that if we only know that 
is divisible by 7, then 
may or may not be a multiple of 7.
Suppose we know both.
If we know that 
is divisible by 7, then, when 
is divided by 7, the remainder is 1, and we can write for some integer 
:
If we know that 
is divisible by 7, then, when 
is divided by 7, the remainder is 6, and we can write for some integer 
:
Then
for some integer quotient 
So, when 
is divided by 7, the remainder is 6, and 
is not a multiple of 7. This makes the two statements together sufficient information.
If we only know that
Example 1: If
Example 2: If
A similar argument shows that if we only know that
Suppose we know both.
If we know that
If we know that
Then
So, when
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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
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
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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.
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.
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What is the last digit of a positive integer 
?
Statement 1: The last digit of 
is 1.
Statement 2: The last digit of 
is 1.
What is the last digit of a positive integer
Statement 1: The last digit of
Statement 2: The last digit of
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If the last digit of 
is 1, then the last digit of 
is either 1 or 9.
If the last digit of 
is 1, however, the last digit of 
must be 1.
If the last digit of
If the last digit of
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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.
This six-digit number has two digits missing:
Statement 1: The number is divisible by 4.
Statement 2: The number is divisible by 3.
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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.
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.
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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.
This five-digit number has two digits missing:
Statement 1: The number is divisible by 5.
Statement 2: The number is not divisible by 3.
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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.
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.
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You are given four numbers, 
. How many of these numbers have value 0?
Statement 1: 
Statement 2: 
You are given four numbers,
Statement 1:
Statement 2:
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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.
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
If you know the second statement, you know that
If you know both statements, though, you know that
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You are given four numbers, 
. You know that exactly one of these numbers has value 0. Which one is it?
Statement 1: 
Statement 2: 
You are given four numbers,
Statement 1:
Statement 2:
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The product of two or more numbers is 0 if and only if at least one of the numbers is 0.
Each of the individual statements eliminates only two possible answers to the question; Statement 1 eliminates 
and 
, and Statement 2 eliminates 
and 
. The two together, however, eliminate all possible answers except for 
, leaving that to be the number worth zero.
The product of two or more numbers is 0 if and only if at least one of the numbers is 0.
Each of the individual statements eliminates only two possible answers to the question; Statement 1 eliminates
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