Integers - GRE Quantitative Reasoning
Card 1 of 1224
Which of the following is a prime number?
Which of the following is a prime number?
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a prime number is divisible by itself and 1 only
list the factors of each number:
6: 1,2,3,6
9: 1,3,9
71: 1,71
51: 1, 3,17,51
15: 1,3,5,15
a prime number is divisible by itself and 1 only
list the factors of each number:
6: 1,2,3,6
9: 1,3,9
71: 1,71
51: 1, 3,17,51
15: 1,3,5,15
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If a is the greatest common divisor of 64 and 14 and b is the least common multiple of 16 and 52 then a + b = ?
If a is the greatest common divisor of 64 and 14 and b is the least common multiple of 16 and 52 then a + b = ?
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The greatest common divisor of 64 and 14 is 2, as found by the prime factorization of 64 and 14. The least common multiple of 16 and 52 is 208, which can be found by looking at the decimal when 52 is divided by 16. The remainder is 0.25, or 1/4 so the fourth multiple of 52 is 208, which is also divisible by 16.
The greatest common divisor of 64 and 14 is 2, as found by the prime factorization of 64 and 14. The least common multiple of 16 and 52 is 208, which can be found by looking at the decimal when 52 is divided by 16. The remainder is 0.25, or 1/4 so the fourth multiple of 52 is 208, which is also divisible by 16.
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A prime number is divisible by:
A prime number is divisible by:
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The definition of a prime number is a number that is divisible by only one and itself. A prime number can't be divided by zero, because numbers divided by zero are undefined. The smallest prime number is 2, which is also the only even prime.
The definition of a prime number is a number that is divisible by only one and itself. A prime number can't be divided by zero, because numbers divided by zero are undefined. The smallest prime number is 2, which is also the only even prime.
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If x is a prime number, then 3_x_ is
If x is a prime number, then 3_x_ is
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Pick a prime number to see that 3_x_ is not always even, for example 3 * 3 = 9.
But 2 is a prime number as well, so 3 * 2 = 6 which is even, so we can't say that 3_x_ is either even or odd.
Neither 9 nor 6 in our above example is prime, so 3_x_ is not a prime number.
Lastly, 9 is not divisible by 4, so 3_x_ is not always divisible by 4.
Therefore the answer is "Cannot be determined".
Pick a prime number to see that 3_x_ is not always even, for example 3 * 3 = 9.
But 2 is a prime number as well, so 3 * 2 = 6 which is even, so we can't say that 3_x_ is either even or odd.
Neither 9 nor 6 in our above example is prime, so 3_x_ is not a prime number.
Lastly, 9 is not divisible by 4, so 3_x_ is not always divisible by 4.
Therefore the answer is "Cannot be determined".
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What is the sum of all of the four-digit integers that can be created with the digits 1, 2, 3, and 4?
What is the sum of all of the four-digit integers that can be created with the digits 1, 2, 3, and 4?
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First we need to find out how many possible numbers there are. The number of possible four-digit numbers with four different digits is simply 4 * 4 * 4 * 4 = 256.
To find the sum, the formula we must remember is sum = average * number of values. The last piece that's missing in this formula is the average. To find this, we can average the first and last number, since the numbers are consecutive. The smallest number that can be created from 1, 2, 3, and 4 is 1111, and the largest number possible is 4444. Then the average is (1111 + 4444)/2.
So sum = 5555/2 * 256 = 711,040.
First we need to find out how many possible numbers there are. The number of possible four-digit numbers with four different digits is simply 4 * 4 * 4 * 4 = 256.
To find the sum, the formula we must remember is sum = average * number of values. The last piece that's missing in this formula is the average. To find this, we can average the first and last number, since the numbers are consecutive. The smallest number that can be created from 1, 2, 3, and 4 is 1111, and the largest number possible is 4444. Then the average is (1111 + 4444)/2.
So sum = 5555/2 * 256 = 711,040.
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Quantity A: The sum of all integers from 49 to 98 inclusive.
Quantity B: The sum of all integers from 51 to 99 inclusive.
Quantity A: The sum of all integers from 49 to 98 inclusive.
Quantity B: The sum of all integers from 51 to 99 inclusive.
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For each quantity, only count the integers that aren't in the other quantity. Both quantities include the numbers 51 to 98, so those numbers won't affect which is greater. Only Quantity A has 49 and 50 (for a total of 99) and only Quantity B has 99. Since the excluded numbers from both quantities equal 99, you can conclude that the 2 quantities are equal.
For each quantity, only count the integers that aren't in the other quantity. Both quantities include the numbers 51 to 98, so those numbers won't affect which is greater. Only Quantity A has 49 and 50 (for a total of 99) and only Quantity B has 99. Since the excluded numbers from both quantities equal 99, you can conclude that the 2 quantities are equal.
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Quantity A: The sum of all integers from 1 to 30
Quantity B: 465
Quantity A: The sum of all integers from 1 to 30
Quantity B: 465
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The sum of all integers from 1 to 30 can be found using the formula
, where 
is 30. In this case, the sum equals 465.
The sum of all integers from 1 to 30 can be found using the formula
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If the product of two distinct integers is 
, which of the following could not represent the sum of those two integers?
If the product of two distinct integers is
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When the product of two numbers is positive, that means that either both numbers were positive, or both numbers were negative.
Now, considering the way 
could be factored:
And of course the cases where both values are negative. For each of these potential factors, the sums are then
Absolute value signs are used to denote that either a sum or it's negative suffices. However, recall that we're told the two integers are distinct!
Due to this, neither 
or 
is an acceptable answers, because both the integers would be equivalent and not distinct.
When the product of two numbers is positive, that means that either both numbers were positive, or both numbers were negative.
Now, considering the way
And of course the cases where both values are negative. For each of these potential factors, the sums are then
Absolute value signs are used to denote that either a sum or it's negative suffices. However, recall that we're told the two integers are distinct!
Due to this, neither
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What is the sum of all the integers between 1 and 69, inclusive?
What is the sum of all the integers between 1 and 69, inclusive?
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The formula here is sum = average value * number of values. Since this is a consecutive series, the average can be found by averaging only the first and last terms: (1 + 69)/2 = 35.
sum = average * number of values = 35 * 69 = 2415
The formula here is sum = average value * number of values. Since this is a consecutive series, the average can be found by averaging only the first and last terms: (1 + 69)/2 = 35.
sum = average * number of values = 35 * 69 = 2415
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The product of two distinct integers is 
. Which of the following is a possible sum of these two integers?
The product of two distinct integers is
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Since 
is negative, it is the product of a positive and negative integer. Consider all of the ways that it could be factored, and the sums these factors would produce:
is the answer choice that matches.
Since
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a, b and c are integers, and a and b are not equivalent.
If ax + bx = c, where c is a prime integer, and a and b are positive integers which of the following is a possible value of x?
a, b and c are integers, and a and b are not equivalent.
If ax + bx = c, where c is a prime integer, and a and b are positive integers which of the following is a possible value of x?
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This question tests basic number properties. Prime numbers are numbers which are divisible only by one and themselves. Answer options '2' and '4' are automatically out, because they will always produce even products with a and b, and the sum of two even products is always even. Since no even number greater than 2 is prime, 2 and 4 cannot be answer options. 3 is tempting, until you remember that the sum of any two multiples of 3 is itself divisible by 3, thereby negating any possible answer for c except 3, which is impossible. There are, however, several possible combinations that work with x = 1. For instance, a = 8 and b = 9 means that 8(1) + 9(1) = 17, which is prime. You only need to find one example to demonstrate that an option works. This eliminates the "None of the other answers" option as well.
This question tests basic number properties. Prime numbers are numbers which are divisible only by one and themselves. Answer options '2' and '4' are automatically out, because they will always produce even products with a and b, and the sum of two even products is always even. Since no even number greater than 2 is prime, 2 and 4 cannot be answer options. 3 is tempting, until you remember that the sum of any two multiples of 3 is itself divisible by 3, thereby negating any possible answer for c except 3, which is impossible. There are, however, several possible combinations that work with x = 1. For instance, a = 8 and b = 9 means that 8(1) + 9(1) = 17, which is prime. You only need to find one example to demonstrate that an option works. This eliminates the "None of the other answers" option as well.
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What is the least common multiple of 
and 
?
What is the least common multiple of
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Factor each of our values into prime factors:
350 = 2 * 52 * 7
6270= 2 * 3 * 5 * 11 * 19
To find the least common multiple, we must choose the larger exponent for each of the prime factors involved. Therefore, we will select 2, 52, and 7 from 350 and 3, 11, and 19 from 6270.
Therefore, our least common multiple is 2 * 3 * 52 * 7 * 11 * 19 = 219,450.
Factor each of our values into prime factors:
350 = 2 * 52 * 7
6270= 2 * 3 * 5 * 11 * 19
To find the least common multiple, we must choose the larger exponent for each of the prime factors involved. Therefore, we will select 2, 52, and 7 from 350 and 3, 11, and 19 from 6270.
Therefore, our least common multiple is 2 * 3 * 52 * 7 * 11 * 19 = 219,450.
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What is the least common multiple of 3, 4x, 5y, 6xy, and 10y?
What is the least common multiple of 3, 4x, 5y, 6xy, and 10y?
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Each of the numbers on the list must be able to "fit" (multiply evenly) into a larger number at the same time. I.e. the largest number (multiple) divided by any of the listed numbers will result in a whole number. For the coefficients, the maximum value is 10, and multiplying the highest two coefficients give us 60. Since 60 is divisible evenly by the lower values, we know that it is the least common multiple for the list. For the variables, both x and y will fit evenly into a theoretical number, "xy". We do not need an exponential version of this multiple as there are no exponents in the original list.
Each of the numbers on the list must be able to "fit" (multiply evenly) into a larger number at the same time. I.e. the largest number (multiple) divided by any of the listed numbers will result in a whole number. For the coefficients, the maximum value is 10, and multiplying the highest two coefficients give us 60. Since 60 is divisible evenly by the lower values, we know that it is the least common multiple for the list. For the variables, both x and y will fit evenly into a theoretical number, "xy". We do not need an exponential version of this multiple as there are no exponents in the original list.
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What is the least common multiple of 45 and 60?
What is the least common multiple of 45 and 60?
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The least common multiple is the smallest number that is a multiple of all the numbers in the group. Let's list some multiples of the two numbers and find the smallest number in common to both.
multiples of 45: 45, 90, 135, 180, 225, 270, ...
multiples of 60: 60, 120, 180, 240, 300, 360, ...
The smallest number in common is 180.
The least common multiple is the smallest number that is a multiple of all the numbers in the group. Let's list some multiples of the two numbers and find the smallest number in common to both.
multiples of 45: 45, 90, 135, 180, 225, 270, ...
multiples of 60: 60, 120, 180, 240, 300, 360, ...
The smallest number in common is 180.
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Find the greatest common factor of 16 and 24.
Find the greatest common factor of 16 and 24.
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First, find all of the factors of each number. Factors are the numbers that, like 16 and 24, can evenly be divided. The factors of 16 are 1, 2, 4, 8, 16. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
Now, to find the greatest common factor, we find the largest number that is on both lists. This number is 8.
First, find all of the factors of each number. Factors are the numbers that, like 16 and 24, can evenly be divided. The factors of 16 are 1, 2, 4, 8, 16. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
Now, to find the greatest common factor, we find the largest number that is on both lists. This number is 8.
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What is the greatest common factor of 
and 
?
What is the greatest common factor of
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To make things easier, note 6930 is divisible by 30:
6930 = 231 * 30 = 3 * 77 * 3 * 2 * 5 = 3 * 7 * 11 * 3 * 2 * 5 = 2 * 32 * 5 * 7 * 11
288 = 2 * 144 = 2 * 12 * 12 = 2 * 2 * 2 * 3 * 2 * 2 * 3 = 25 * 32
Consider each of these "next to each other":
25 * 32
2 * 32 * 5 * 7 * 11
Each shares factors of 2 and 3. In the case of 2, they share 1 factor. In the case of 3, they share 2 factors. Therefore, their greatest common factor is: 2 * 32 = 2 * 9 = 18
To make things easier, note 6930 is divisible by 30:
6930 = 231 * 30 = 3 * 77 * 3 * 2 * 5 = 3 * 7 * 11 * 3 * 2 * 5 = 2 * 32 * 5 * 7 * 11
288 = 2 * 144 = 2 * 12 * 12 = 2 * 2 * 2 * 3 * 2 * 2 * 3 = 25 * 32
Consider each of these "next to each other":
25 * 32
2 * 32 * 5 * 7 * 11
Each shares factors of 2 and 3. In the case of 2, they share 1 factor. In the case of 3, they share 2 factors. Therefore, their greatest common factor is: 2 * 32 = 2 * 9 = 18
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What is the greatest common factor of 18 and 24?
What is the greatest common factor of 18 and 24?
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The greatest common factor is the greatest factor that divides both numbers. To find the greatest common factor, first list the prime factors of each number.
18 = 2 * 3 * 3
24 = 2 * 2 * 2 * 3
18 and 24 share one 2 and one 3 in common. We multiply them to get the GCF, so 2 * 3 = 6 is the GCF of 18 and 24.
The greatest common factor is the greatest factor that divides both numbers. To find the greatest common factor, first list the prime factors of each number.
18 = 2 * 3 * 3
24 = 2 * 2 * 2 * 3
18 and 24 share one 2 and one 3 in common. We multiply them to get the GCF, so 2 * 3 = 6 is the GCF of 18 and 24.
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What is half of the third smallest prime number multiplied by the smallest two digit prime number?
What is half of the third smallest prime number multiplied by the smallest two digit prime number?
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The third smallest prime number is 5. (Don't forget that 2 is a prime number, but 1 is not!)
The smallest two digit prime number is 11.
Now we can evaluate the entire expression:
The third smallest prime number is 5. (Don't forget that 2 is a prime number, but 1 is not!)
The smallest two digit prime number is 11.
Now we can evaluate the entire expression:
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Which of the following pairs of numbers are twin primes?
Which of the following pairs of numbers are twin primes?
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For starters, 1 is not a prime number, so eliminate the answer choices with 1 in them. Even if you have no idea what twin primes are, at least you've narrowed down the possibilities.
Twin primes are consecutive prime numbers with one even number in between them. 3 and 5 is the only set of twin primes listed. 2 and 3 are not separated by any numbers, and 13 and 19 are not consecutive primes, nor are they separated by one even number only. You should do your best to remember definitions and formulas such as this one, because these questions are considered "free" points on the test. There is no real math involved, just something to remember! Being able to answer a question like this quickly will give you more time for the computationally advanced problems.
For starters, 1 is not a prime number, so eliminate the answer choices with 1 in them. Even if you have no idea what twin primes are, at least you've narrowed down the possibilities.
Twin primes are consecutive prime numbers with one even number in between them. 3 and 5 is the only set of twin primes listed. 2 and 3 are not separated by any numbers, and 13 and 19 are not consecutive primes, nor are they separated by one even number only. You should do your best to remember definitions and formulas such as this one, because these questions are considered "free" points on the test. There is no real math involved, just something to remember! Being able to answer a question like this quickly will give you more time for the computationally advanced problems.
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Quantitative Comparison
Quantity A: The number of prime numbers between 0 and 100, inclusive.
Quantity B: The number of prime numbers between 101 and 200, inclusive.
Quantitative Comparison
Quantity A: The number of prime numbers between 0 and 100, inclusive.
Quantity B: The number of prime numbers between 101 and 200, inclusive.
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As we go up on the number line, the number of primes decreases almost exponentially. Therefore there are far more prime numbers between 0 and 100 than there are between 101 and 200. This is a general number theory point that is important to know, but trying to come up with some primes in these two groups will also quickly demonstrate this principle.
As we go up on the number line, the number of primes decreases almost exponentially. Therefore there are far more prime numbers between 0 and 100 than there are between 101 and 200. This is a general number theory point that is important to know, but trying to come up with some primes in these two groups will also quickly demonstrate this principle.
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