Properties of Integers - GMAT Quantitative
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What is the sum of the prime numbers that are greater than 50 but less than 60?
What is the sum of the prime numbers that are greater than 50 but less than 60?
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A prime number is only divisible by the number 1 and itself. Of the integers between 50 and 60, all of the even integers are also divisible by the number 2 so they are not prime numbers. The integers 51 and 57 are divisible by 3. The integer 55 is divisible by 5. The only integers that are prime are 53 and 59. The sum of these two integers is 112.
A prime number is only divisible by the number 1 and itself. Of the integers between 50 and 60, all of the even integers are also divisible by the number 2 so they are not prime numbers. The integers 51 and 57 are divisible by 3. The integer 55 is divisible by 5. The only integers that are prime are 53 and 59. The sum of these two integers is 112.
What is the product of the four smallest prime numbers?
What is the product of the four smallest prime numbers?
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We must remember that 0 and 1 are NOT prime numbers, but 2 is.
The four smallest prime numbers are 2, 3, 5, and 7. Then 2 * 3 * 5 * 7 = 210.
Note: There are NO negative prime numbers, so we don't have to look for tiny, negative numbers here.
We must remember that 0 and 1 are NOT prime numbers, but 2 is.
The four smallest prime numbers are 2, 3, 5, and 7. Then 2 * 3 * 5 * 7 = 210.
Note: There are NO negative prime numbers, so we don't have to look for tiny, negative numbers here.
If a and b are even integers, what must be odd?
If a and b are even integers, what must be odd?
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The sum (or difference) or 2 even integers is even. Similarly, the product (or quotient) of 2 even integers is also even; therefore the answer must be a+b-1, which can be easily checked by plugging in any two even numbers.
For example, if a=2 and b=4, a+b-1=2+4-1=5, which is odd.
The sum (or difference) or 2 even integers is even. Similarly, the product (or quotient) of 2 even integers is also even; therefore the answer must be a+b-1, which can be easily checked by plugging in any two even numbers.
For example, if a=2 and b=4, a+b-1=2+4-1=5, which is odd.
Three consecutive numbers add up to 36. What is the smallest number?
Three consecutive numbers add up to 36. What is the smallest number?
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The sum of 3 consecutive numbers would be
which simplifies into
Set that equation equal to 36 and solve.
The sum of 3 consecutive numbers would be
which simplifies into
Set that equation equal to 36 and solve.
Becky has to choose from 4 pairs of pants, 6 shirts, and 2 pairs of shoes for an interview. If an outfit consists of 1 pair of pants, 1 pair of shoes, and 1 shirt, how many options does she have?
Becky has to choose from 4 pairs of pants, 6 shirts, and 2 pairs of shoes for an interview. If an outfit consists of 1 pair of pants, 1 pair of shoes, and 1 shirt, how many options does she have?
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To find total number, multiply the number of each item.
To find total number, multiply the number of each item.
What is the greatest prime factor of 
What is the greatest prime factor of
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The only prime factors are 3 and 5, therefore, 5 will be the greatest prime factor.
The only prime factors are 3 and 5, therefore, 5 will be the greatest prime factor.
Solve:
Solve:
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The sum is 8,053,467
The sum is 8,053,467
For how many integers, 
, is 
an integer?
For how many integers,
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Since the numerator is always 1 greater than the denominator, we know that for large enough values of 
, it's never going to be an integer (one will be even, other odd). In fact, there are only 2 cases where this can be done. The first is dividing 0. Since 0 is divisible by every number, if the numerator is 0, then we will still get an integer. Thus one answer is 
The other answer occurs as a special case as well. We can divide any number by 1 evenly, so when the denominator is 1 we get an integer:
In every other case, we will have a non-integer.
Since the numerator is always 1 greater than the denominator, we know that for large enough values of
The other answer occurs as a special case as well. We can divide any number by 1 evenly, so when the denominator is 1 we get an integer:
In every other case, we will have a non-integer.
A positive integer 
divided by 9 yields remainder 7. What remainder does 
divided by 9 yield?
A positive integer
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The easiest way to explain this is by example.
We can use 16, since 
This makes 4 the correct choice.
The easiest way to explain this is by example.
We can use 16, since
This makes 4 the correct choice.
If a positive integer 
is divided by another positive integer 
, then the quotient is 6, and there is no remainder.
Which of these choices is a possible value of 
?
If a positive integer
Which of these choices is a possible value of
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The conditions of the problem can be rewritten as 
, or 
.
, meaning that the sum of the two numbers is a multiple of 7. Each of the given choices is a multiple of 7, so any of them can be 
.
The conditions of the problem can be rewritten as
A positive integer 
divided by 5 yields remainder 3. What remainder does 
divided by 5 yield?
A positive integer
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Any integer that is divisible by 5 must end in a 0 or a 5, so any integer which, when divided by 5, yields remainder 3 must end in 3 or 8. Since the cubes of 3 and 8 are 27 and 512, the cube of an integer that ends in one of these digits must end in 2 or 7, meaning that, when divided by 5, the remainder will be 2.
Any integer that is divisible by 5 must end in a 0 or a 5, so any integer which, when divided by 5, yields remainder 3 must end in 3 or 8. Since the cubes of 3 and 8 are 27 and 512, the cube of an integer that ends in one of these digits must end in 2 or 7, meaning that, when divided by 5, the remainder will be 2.
What is the first digit in the base-six representation of the number 936?
What is the first digit in the base-six representation of the number 936?
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, or, equivalently, 
. This makes 936 a four-digit number when written in base six. The first digit is equal to the number of times 216 divides into 936.
,
4 is the first digit of this number.
4 is the first digit of this number.
What is the last digit in the base-eight representation of the number 735?
What is the last digit in the base-eight representation of the number 735?
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Divide 735 by 8. The remainder is the last digit of the base-eight representation.
The correct choice is 7.
Divide 735 by 8. The remainder is the last digit of the base-eight representation.
The correct choice is 7.
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?
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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.
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?
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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.
A one-hundred-digit integer that is divisible by 6 begins with a 1; its next ninety-seven digits are all zeroes. Which of the following could be its last two digits?
A one-hundred-digit integer that is divisible by 6 begins with a 1; its next ninety-seven digits are all zeroes. Which of the following could be its last two digits?
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For a number to be divisible by 6, it must be both even and a multiple of 3. None of the choices can be eliminated by that first criterion. But to fit the second criterion, the digit sum must be divisible by 3. We need only add the nonzero digits of each choice, remembering to include the leading 1 in the sum:
Of these choices, only 56 passes the divisibility test, so it is the correct choice.
For a number to be divisible by 6, it must be both even and a multiple of 3. None of the choices can be eliminated by that first criterion. But to fit the second criterion, the digit sum must be divisible by 3. We need only add the nonzero digits of each choice, remembering to include the leading 1 in the sum:
Of these choices, only 56 passes the divisibility test, so it is the correct choice.
Rewrite 117 as a number in base eight.
Rewrite 117 as a number in base eight.
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One way to do this:
Divide 117 by 8. The remainder will be the last digit.
Now divide the quotient by 8. This remainder will be the second-to-last digit.
The quotient is less than 8, so it will be the first digit. The base-eight equivalent of 117 is 
One way to do this:
Divide 117 by 8. The remainder will be the last digit.
Now divide the quotient by 8. This remainder will be the second-to-last digit.
The quotient is less than 8, so it will be the first digit. The base-eight equivalent of 117 is
Express the quotient in scientific notation:
Express the quotient in scientific notation:
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All of these expressions are equivalent to the correct quotient. But a number in scientific notation takes the form
, where 
Only 
takes this form.
All of these expressions are equivalent to the correct quotient. But a number in scientific notation takes the form
Only
Which of the following numbers has a five in the last digit of its base-six representation?
Which of the following numbers has a five in the last digit of its base-six representation?
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If a number has a 5 in its base-six representation, then the number divided by 6 yields a remainder of 5. We test each of these five numbers by dividing each by 6 and noting the remainder:
Among the five choices, only 743 yields remainder 5 when divided by 6, so this is the correct choice.
If a number has a 5 in its base-six representation, then the number divided by 6 yields a remainder of 5. We test each of these five numbers by dividing each by 6 and noting the remainder:
Among the five choices, only 743 yields remainder 5 when divided by 6, so this is the correct choice.
What is the product of all of the prime numbers between 20 and 30?
What is the product of all of the prime numbers between 20 and 30?
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There are two prime numbers between 20 and 30; they are 23 and 29. Their product:
There are two prime numbers between 20 and 30; they are 23 and 29. Their product: