Integers - GRE Quantitative Reasoning
Card 0 of 1224
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?
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.
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
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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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?
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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If the sum of four consecutive numbers is 38, what is the mean of the largest and the smallest of the four numbers?
If the sum of four consecutive numbers is 38, what is the mean of the largest and the smallest of the four numbers?
The sum of 4 consecutive numbers being 38 can be written as the following equation.
We can simplify this to solve for x.
This tells us that the smallest number is 8 and the largest is (8 + 3) = 11. From this, we can find their mean.
The sum of 4 consecutive numbers being 38 can be written as the following equation.
We can simplify this to solve for x.
This tells us that the smallest number is 8 and the largest is (8 + 3) = 11. From this, we can find their mean.
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The average of five consecutive integers is 
. What is the largest of these integers?
The average of five consecutive integers is
There are two ways to figure out this list of integers. On the one hand, you might know that the average of a set of consecutive integers is the "middle value" of that set. So, if the average is 
and the size 
, the list must be:
Another way to figure this out is to represent your integers as:
The average of these values will be all of these numbers added together and then divided by 
. This gives us:
Multiply both sides by 
:
Finish solving:
This means that the largest value is:
There are two ways to figure out this list of integers. On the one hand, you might know that the average of a set of consecutive integers is the "middle value" of that set. So, if the average is
Another way to figure this out is to represent your integers as:
The average of these values will be all of these numbers added together and then divided by
Multiply both sides by
Finish solving:
This means that the largest value is:
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The sum of a set of 
consecutive odd integers is 
. What is the fourth number in this set?
The sum of a set of
We can represent our numbers as:
will have to be an odd number since the whole sequence is odd. However, this will work out when we do the math. Now, we know that all of these added up will be 
. We have 
and the sum of the set 
, the sum of which is 
.
Thus, we know:
Solve for 
:
Therefore, the fourth element will be 
:
We can represent our numbers as:
Thus, we know:
Solve for
Therefore, the fourth element will be
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Solve for 
:
Solve for
To solve this problem, you need to get your variable isolated on one side of the equation:
Taking this step will elminate the 
on the side with 
:
Now divide by 
to solve for 
:
The important step here is to be able to add the negative numbers. When you add negative numbers, they create lower negative numbers (if you prefer to think about it another way, you can think of adding negative numbers as subtracting one of the negative numbers from the other).
To solve this problem, you need to get your variable isolated on one side of the equation:
Taking this step will elminate the
Now divide by
The important step here is to be able to add the negative numbers. When you add negative numbers, they create lower negative numbers (if you prefer to think about it another way, you can think of adding negative numbers as subtracting one of the negative numbers from the other).
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Simplify 
Simplify
The answer is 
Make sure to distribute negatives throughout the second half of the equation.
The answer is
Make sure to distribute negatives throughout the second half of the equation.
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Solve for 
:
Solve for
To solve this problem, first you must add 
to both sides of the problem. This will yield a result on the right side of the equation of 
, because a negative number added to a negative number will create a lower number (i.e. further away from zero, and still negative). Then you divide both sides by two, and you are left with 
.
To solve this problem, first you must add
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The product of two consecutive positive integers is 272. What is the larger of the two integers?
The product of two consecutive positive integers is 272. What is the larger of the two integers?
In order to multiply to 272, the units digits of the two integers will have to multiply to a number with a units digit of 2. For 17, you can see that 6 x 7 (the units digits of 16 and 17) = 42. The best strategy here is to plug in the choices using the units digit strategy.
In order to multiply to 272, the units digits of the two integers will have to multiply to a number with a units digit of 2. For 17, you can see that 6 x 7 (the units digits of 16 and 17) = 42. The best strategy here is to plug in the choices using the units digit strategy.
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What is the units digit of 
?
What is the units digit of
The units digit of any product depends on the units digits of the 2 numbers multiplied, which in this case is 3 and 4. Since 
, the units digit of 
is 2.
The units digit of any product depends on the units digits of the 2 numbers multiplied, which in this case is 3 and 4. Since
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What is the units digit of
What is the units digit of
The units digit of the product of any two numbers is the same as the units digit of the product of the two numbers' units digits. In this case, it would be the units digit of 
, which is 
.
The units digit of the product of any two numbers is the same as the units digit of the product of the two numbers' units digits. In this case, it would be the units digit of
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Choose the answer below which best solves the following equation:
Choose the answer below which best solves the following equation:
When multiplying integers, if one of the integers is negative, your answer will be negative:
First multiply the ones digits together.
Next multiply the ones digit of the smaller number with the tens digit of the larger number to get the tens digit of the product.
Combining these two and remembering the negative sign we get our final answer:
When multiplying integers, if one of the integers is negative, your answer will be negative:
First multiply the ones digits together.
Next multiply the ones digit of the smaller number with the tens digit of the larger number to get the tens digit of the product.
Combining these two and remembering the negative sign we get our final answer:
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Choose the answer which best solves the following equation:
Choose the answer which best solves the following equation:
When multiplying integers, if both of the signs of the integers are the same (positive, or negative) then your result will be positive:
First multiply the ones digit of both numbers together. If the product is greater than ten remember to carry the one to the tens place.
Next multiply the ones digit of the smaller number with the tens digit of the larger number and add the number that was carried over.
Combine these two together to get:
When multiplying integers, if both of the signs of the integers are the same (positive, or negative) then your result will be positive:
First multiply the ones digit of both numbers together. If the product is greater than ten remember to carry the one to the tens place.
Next multiply the ones digit of the smaller number with the tens digit of the larger number and add the number that was carried over.
Combine these two together to get:
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Which of the following is a prime number?
Which of the following is a prime number?
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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A prime number is divisible by:
A prime number is divisible by:
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
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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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?
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 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?
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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