Properties of Integers - GMAT Quantitative
Card 1 of 624
What is the sum of the odd numbers from 
to 
, inclusive?
What is the sum of the odd numbers from
Tap to reveal answer
We can divide this group number into two parts. The first part is 
and the second part is 
.
The first group is symmetrical, so the sum of this group of numbers is 0. Now for the second part there are only two numbers left according to the question (sum of odd numbers), which is 157 and 159.
Therefore, the answer is 
.
We can divide this group number into two parts. The first part is
The first group is symmetrical, so the sum of this group of numbers is 0. Now for the second part there are only two numbers left according to the question (sum of odd numbers), which is 157 and 159.
Therefore, the answer is
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How many integers from 71 to 100 inclusive do not have 2, 3, or 5 as a factor?
How many integers from 71 to 100 inclusive do not have 2, 3, or 5 as a factor?
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The factors of 2 are exactly the integers that end in 2, 4, 6, 8, or 0, and the factors of 5 are exactly the integers that end in 5 or 0. Therefore, we will start with the integers that end in 1, 3, 7, and 9, and eliminate the multiples of 3:
Of these, 81, 87, 93, and 99 are the only multiples of 3. Remove them and we have the set
,
a set of eight elements.
The factors of 2 are exactly the integers that end in 2, 4, 6, 8, or 0, and the factors of 5 are exactly the integers that end in 5 or 0. Therefore, we will start with the integers that end in 1, 3, 7, and 9, and eliminate the multiples of 3:
Of these, 81, 87, 93, and 99 are the only multiples of 3. Remove them and we have the set
a set of eight elements.
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What is the sum of the odd numbers from to , inclusive?
What is the sum of the odd numbers from
Tap to reveal answer
We can divide this group number into two parts. The first part is and the second part is .
The first group is symmetrical, so the sum of this group of numbers is 0. Now for the second part there are only two numbers left according to the question (sum of odd numbers), which is 157 and 159.
Therefore, the answer is .
We can divide this group number into two parts. The first part is
The first group is symmetrical, so the sum of this group of numbers is 0. Now for the second part there are only two numbers left according to the question (sum of odd numbers), which is 157 and 159.
Therefore, the answer is
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How many integers from 71 to 100 inclusive do not have 2, 3, or 5 as a factor?
How many integers from 71 to 100 inclusive do not have 2, 3, or 5 as a factor?
Tap to reveal answer
The factors of 2 are exactly the integers that end in 2, 4, 6, 8, or 0, and the factors of 5 are exactly the integers that end in 5 or 0. Therefore, we will start with the integers that end in 1, 3, 7, and 9, and eliminate the multiples of 3:
Of these, 81, 87, 93, and 99 are the only multiples of 3. Remove them and we have the set
,
a set of eight elements.
The factors of 2 are exactly the integers that end in 2, 4, 6, 8, or 0, and the factors of 5 are exactly the integers that end in 5 or 0. Therefore, we will start with the integers that end in 1, 3, 7, and 9, and eliminate the multiples of 3:
Of these, 81, 87, 93, and 99 are the only multiples of 3. Remove them and we have the set
a set of eight elements.
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What is the sum of the odd numbers from to , inclusive?
What is the sum of the odd numbers from
Tap to reveal answer
We can divide this group number into two parts. The first part is and the second part is .
The first group is symmetrical, so the sum of this group of numbers is 0. Now for the second part there are only two numbers left according to the question (sum of odd numbers), which is 157 and 159.
Therefore, the answer is .
We can divide this group number into two parts. The first part is
The first group is symmetrical, so the sum of this group of numbers is 0. Now for the second part there are only two numbers left according to the question (sum of odd numbers), which is 157 and 159.
Therefore, the answer is
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How many integers from 71 to 100 inclusive do not have 2, 3, or 5 as a factor?
How many integers from 71 to 100 inclusive do not have 2, 3, or 5 as a factor?
Tap to reveal answer
The factors of 2 are exactly the integers that end in 2, 4, 6, 8, or 0, and the factors of 5 are exactly the integers that end in 5 or 0. Therefore, we will start with the integers that end in 1, 3, 7, and 9, and eliminate the multiples of 3:
Of these, 81, 87, 93, and 99 are the only multiples of 3. Remove them and we have the set
,
a set of eight elements.
The factors of 2 are exactly the integers that end in 2, 4, 6, 8, or 0, and the factors of 5 are exactly the integers that end in 5 or 0. Therefore, we will start with the integers that end in 1, 3, 7, and 9, and eliminate the multiples of 3:
Of these, 81, 87, 93, and 99 are the only multiples of 3. Remove them and we have the set
a set of eight elements.
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What is the sum of the odd numbers from to , inclusive?
What is the sum of the odd numbers from
Tap to reveal answer
We can divide this group number into two parts. The first part is and the second part is .
The first group is symmetrical, so the sum of this group of numbers is 0. Now for the second part there are only two numbers left according to the question (sum of odd numbers), which is 157 and 159.
Therefore, the answer is .
We can divide this group number into two parts. The first part is
The first group is symmetrical, so the sum of this group of numbers is 0. Now for the second part there are only two numbers left according to the question (sum of odd numbers), which is 157 and 159.
Therefore, the answer is
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How many integers from 71 to 100 inclusive do not have 2, 3, or 5 as a factor?
How many integers from 71 to 100 inclusive do not have 2, 3, or 5 as a factor?
Tap to reveal answer
The factors of 2 are exactly the integers that end in 2, 4, 6, 8, or 0, and the factors of 5 are exactly the integers that end in 5 or 0. Therefore, we will start with the integers that end in 1, 3, 7, and 9, and eliminate the multiples of 3:
Of these, 81, 87, 93, and 99 are the only multiples of 3. Remove them and we have the set
,
a set of eight elements.
The factors of 2 are exactly the integers that end in 2, 4, 6, 8, or 0, and the factors of 5 are exactly the integers that end in 5 or 0. Therefore, we will start with the integers that end in 1, 3, 7, and 9, and eliminate the multiples of 3:
Of these, 81, 87, 93, and 99 are the only multiples of 3. Remove them and we have the set
a set of eight elements.
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What is the sum of the odd numbers from to , inclusive?
What is the sum of the odd numbers from
Tap to reveal answer
We can divide this group number into two parts. The first part is and the second part is .
The first group is symmetrical, so the sum of this group of numbers is 0. Now for the second part there are only two numbers left according to the question (sum of odd numbers), which is 157 and 159.
Therefore, the answer is .
We can divide this group number into two parts. The first part is
The first group is symmetrical, so the sum of this group of numbers is 0. Now for the second part there are only two numbers left according to the question (sum of odd numbers), which is 157 and 159.
Therefore, the answer is
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How many integers from 71 to 100 inclusive do not have 2, 3, or 5 as a factor?
How many integers from 71 to 100 inclusive do not have 2, 3, or 5 as a factor?
Tap to reveal answer
The factors of 2 are exactly the integers that end in 2, 4, 6, 8, or 0, and the factors of 5 are exactly the integers that end in 5 or 0. Therefore, we will start with the integers that end in 1, 3, 7, and 9, and eliminate the multiples of 3:
Of these, 81, 87, 93, and 99 are the only multiples of 3. Remove them and we have the set
,
a set of eight elements.
The factors of 2 are exactly the integers that end in 2, 4, 6, 8, or 0, and the factors of 5 are exactly the integers that end in 5 or 0. Therefore, we will start with the integers that end in 1, 3, 7, and 9, and eliminate the multiples of 3:
Of these, 81, 87, 93, and 99 are the only multiples of 3. Remove them and we have the set
a set of eight elements.
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What is the sum of the odd numbers from to , inclusive?
What is the sum of the odd numbers from
Tap to reveal answer
We can divide this group number into two parts. The first part is and the second part is .
The first group is symmetrical, so the sum of this group of numbers is 0. Now for the second part there are only two numbers left according to the question (sum of odd numbers), which is 157 and 159.
Therefore, the answer is .
We can divide this group number into two parts. The first part is
The first group is symmetrical, so the sum of this group of numbers is 0. Now for the second part there are only two numbers left according to the question (sum of odd numbers), which is 157 and 159.
Therefore, the answer is
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How many integers from 71 to 100 inclusive do not have 2, 3, or 5 as a factor?
How many integers from 71 to 100 inclusive do not have 2, 3, or 5 as a factor?
Tap to reveal answer
The factors of 2 are exactly the integers that end in 2, 4, 6, 8, or 0, and the factors of 5 are exactly the integers that end in 5 or 0. Therefore, we will start with the integers that end in 1, 3, 7, and 9, and eliminate the multiples of 3:
Of these, 81, 87, 93, and 99 are the only multiples of 3. Remove them and we have the set
,
a set of eight elements.
The factors of 2 are exactly the integers that end in 2, 4, 6, 8, or 0, and the factors of 5 are exactly the integers that end in 5 or 0. Therefore, we will start with the integers that end in 1, 3, 7, and 9, and eliminate the multiples of 3:
Of these, 81, 87, 93, and 99 are the only multiples of 3. Remove them and we have the set
a set of eight elements.
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What is the sum of the odd numbers from to , inclusive?
What is the sum of the odd numbers from
Tap to reveal answer
We can divide this group number into two parts. The first part is and the second part is .
The first group is symmetrical, so the sum of this group of numbers is 0. Now for the second part there are only two numbers left according to the question (sum of odd numbers), which is 157 and 159.
Therefore, the answer is .
We can divide this group number into two parts. The first part is
The first group is symmetrical, so the sum of this group of numbers is 0. Now for the second part there are only two numbers left according to the question (sum of odd numbers), which is 157 and 159.
Therefore, the answer is
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How many integers from 71 to 100 inclusive do not have 2, 3, or 5 as a factor?
How many integers from 71 to 100 inclusive do not have 2, 3, or 5 as a factor?
Tap to reveal answer
The factors of 2 are exactly the integers that end in 2, 4, 6, 8, or 0, and the factors of 5 are exactly the integers that end in 5 or 0. Therefore, we will start with the integers that end in 1, 3, 7, and 9, and eliminate the multiples of 3:
Of these, 81, 87, 93, and 99 are the only multiples of 3. Remove them and we have the set
,
a set of eight elements.
The factors of 2 are exactly the integers that end in 2, 4, 6, 8, or 0, and the factors of 5 are exactly the integers that end in 5 or 0. Therefore, we will start with the integers that end in 1, 3, 7, and 9, and eliminate the multiples of 3:
Of these, 81, 87, 93, and 99 are the only multiples of 3. Remove them and we have the set
a set of eight elements.
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What is the sum of the odd numbers from to , inclusive?
What is the sum of the odd numbers from
Tap to reveal answer
We can divide this group number into two parts. The first part is and the second part is .
The first group is symmetrical, so the sum of this group of numbers is 0. Now for the second part there are only two numbers left according to the question (sum of odd numbers), which is 157 and 159.
Therefore, the answer is .
We can divide this group number into two parts. The first part is
The first group is symmetrical, so the sum of this group of numbers is 0. Now for the second part there are only two numbers left according to the question (sum of odd numbers), which is 157 and 159.
Therefore, the answer is
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How many integers from 71 to 100 inclusive do not have 2, 3, or 5 as a factor?
How many integers from 71 to 100 inclusive do not have 2, 3, or 5 as a factor?
Tap to reveal answer
The factors of 2 are exactly the integers that end in 2, 4, 6, 8, or 0, and the factors of 5 are exactly the integers that end in 5 or 0. Therefore, we will start with the integers that end in 1, 3, 7, and 9, and eliminate the multiples of 3:
Of these, 81, 87, 93, and 99 are the only multiples of 3. Remove them and we have the set
,
a set of eight elements.
The factors of 2 are exactly the integers that end in 2, 4, 6, 8, or 0, and the factors of 5 are exactly the integers that end in 5 or 0. Therefore, we will start with the integers that end in 1, 3, 7, and 9, and eliminate the multiples of 3:
Of these, 81, 87, 93, and 99 are the only multiples of 3. Remove them and we have the set
a set of eight elements.
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Add the composite numbers between 81 and 90 inclusive.
Add the composite numbers between 81 and 90 inclusive.
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The only prime numbers from 81 to 90 are 83 and 89, so we add:
The only prime numbers from 81 to 90 are 83 and 89, so we add:
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Add the factors of 29.
Add the factors of 29.
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29 is a prime number and therefore has two factors, 1 and itself. The sum of the two is 30.
29 is a prime number and therefore has two factors, 1 and itself. The sum of the two is 30.
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You are given that the product of eight nonzero numbers is negative. Which of the following is not possible?
You are given that the product of eight nonzero numbers is negative. Which of the following is not possible?
Tap to reveal answer
The product of a group of nonzero numbers is negative if and only if an odd number of these factors is negative. This occurs in each of these scenarios except for one - all of the numbers (eight) being negative.
The product of a group of nonzero numbers is negative if and only if an odd number of these factors is negative. This occurs in each of these scenarios except for one - all of the numbers (eight) being negative.
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You are given that the product of eight numbers, each of which is nonzero, is positive. Which of the following is not possible?
You are given that the product of eight numbers, each of which is nonzero, is positive. Which of the following is not possible?
Tap to reveal answer
The product of a group of nonzero numbers is positive if and only if an even number of these factors is negative. This occurs in each of these scenarios.
The product of a group of nonzero numbers is positive if and only if an even number of these factors is negative. This occurs in each of these scenarios.
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