How to factor a number - GRE Quantitative Reasoning
Card 0 of 96
What is the largest possible integer value of 
if 
divides 16! evenly?
What is the largest possible integer value of
This question is really asking, “How many factors of 4 are there in 16!”? To ascertain this, list all the even numbers and count the total number of 2s among those factors.
Respectively, 16, 14, 12, 10, 8, 6, 4, 2 have 4, 1, 2, 1, 3, 1, 2, 1 factors of 2.
The total then is 15. This means that you have a factor of 215, which is the same as 47 * 2; therefore, since you are asked for the largest integer value of n, 7 is your answer.
Any larger integer value would not allow 4n to divide 16! evenly.
This question is really asking, “How many factors of 4 are there in 16!”? To ascertain this, list all the even numbers and count the total number of 2s among those factors.
Respectively, 16, 14, 12, 10, 8, 6, 4, 2 have 4, 1, 2, 1, 3, 1, 2, 1 factors of 2.
The total then is 15. This means that you have a factor of 215, which is the same as 47 * 2; therefore, since you are asked for the largest integer value of n, 7 is your answer.
Any larger integer value would not allow 4n to divide 16! evenly.
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Column A
5!/3!
Column B
6!/4!
Column A
5!/3!
Column B
6!/4!
This is a basic factorial question. A factorial is equal to the number times every positive whole number smaller than itself. In Column A, the numerator is 5 * 4 * 3 * 2 * 1 while the denominator is 3 * 2 * 1.
As you can see, the 3 * 2 * 1 can be cancelled out from both the numerator and denominator, leaving only 5 * 4.
The value for Column A is 5 * 4 = 20.
In Column B, the numerator is 6 * 5 * 4 * 3 * 2 * 1 while the denominator is 4 * 3 * 2 * 1. After simplifying, Column B gives a value of 6 * 5, or 30.
Thus, Column B is greater than Column A.
This is a basic factorial question. A factorial is equal to the number times every positive whole number smaller than itself. In Column A, the numerator is 5 * 4 * 3 * 2 * 1 while the denominator is 3 * 2 * 1.
As you can see, the 3 * 2 * 1 can be cancelled out from both the numerator and denominator, leaving only 5 * 4.
The value for Column A is 5 * 4 = 20.
In Column B, the numerator is 6 * 5 * 4 * 3 * 2 * 1 while the denominator is 4 * 3 * 2 * 1. After simplifying, Column B gives a value of 6 * 5, or 30.
Thus, Column B is greater than Column A.
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The prime factorization of 60 is?
The prime factorization of 60 is?
Prime numbers are numbers that can only divided by one and themselves. Breaking 60 into its prime factors yields:
Prime numbers are numbers that can only divided by one and themselves. Breaking 60 into its prime factors yields:
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Which of the following integers are factors of both 24 and 42?
Which of the following integers are factors of both 24 and 42?
3 is the only answer that is a factor of both 24 and 42. 42/3 = 14 and 24/3 = 8. The other answers are either a factor of 24 OR 42 or neither, but not both.
3 is the only answer that is a factor of both 24 and 42. 42/3 = 14 and 24/3 = 8. The other answers are either a factor of 24 OR 42 or neither, but not both.
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721(413) + 211(721) is equal to which of the following?
721(413) + 211(721) is equal to which of the following?
The answer is 721(413 + 211) because we can pull out a common factor, or 721, from both sides of the equation.
The answer is 721(413 + 211) because we can pull out a common factor, or 721, from both sides of the equation.
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n is a positive integer . p = 4 * 6 * 11 * n
Quantity A: The remainder when p is divided by 5
Quantity B: The remainder when p is divided by 33
n is a positive integer . p = 4 * 6 * 11 * n
Quantity A: The remainder when p is divided by 5
Quantity B: The remainder when p is divided by 33
Let's consider Quantity B first. What will the remainder be when p is divided by 33?
4, 6 and 11 are factors of p which means that 2 * 2 * 2 * 3 * 11 * n will equal p. We can group the 3 and 11 to see that 33 will always be a factor of p and will have no remainder. Thus Quantity B will always equal 0 no matter the value of n.
Now consider Quantity A. Let's consider first the values for p when n equals 1 through 5. When n = 1, p = 264, and the remainder is 4/5 or 0.8.
n = 2, p = 528, and the remainder is 3/5 or 0.6.
n = 3, p = 792, and the remainder is 2/5 or 0.4.
n = 4, p = 1056, and the remainder is 1/5 or 0.2.
n = 5, p = 1320, and the remainder is 0 (because when n = 5, 5 becomes a factor of p and thus there is no remainder.
Because Quantity A can be equal to or greater than B, there is not enough information given to determine the relationship.
Let's consider Quantity B first. What will the remainder be when p is divided by 33?
4, 6 and 11 are factors of p which means that 2 * 2 * 2 * 3 * 11 * n will equal p. We can group the 3 and 11 to see that 33 will always be a factor of p and will have no remainder. Thus Quantity B will always equal 0 no matter the value of n.
Now consider Quantity A. Let's consider first the values for p when n equals 1 through 5. When n = 1, p = 264, and the remainder is 4/5 or 0.8.
n = 2, p = 528, and the remainder is 3/5 or 0.6.
n = 3, p = 792, and the remainder is 2/5 or 0.4.
n = 4, p = 1056, and the remainder is 1/5 or 0.2.
n = 5, p = 1320, and the remainder is 0 (because when n = 5, 5 becomes a factor of p and thus there is no remainder.
Because Quantity A can be equal to or greater than B, there is not enough information given to determine the relationship.
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Quantitative Comparison
Quantity A: number of 2's in the prime factorization of 32
Quantity B: number of 2's in the prime factorization of 60
Quantitative Comparison
Quantity A: number of 2's in the prime factorization of 32
Quantity B: number of 2's in the prime factorization of 60
32 = 2 * 16 = 2 * 4 * 4 = 2 * 2 * 2 * 2 * 2 = 25, so Quantity A = 5.
60 = 2 * 30 = 2 * 6 * 5 = 2 * 2 * 3 * 5 = 22 * 3 * 5, so Quantity B = 2.
Quantity A is greater. Even though 60 is a larger number than 32, 32 has more 2's in its prime factorization.
32 = 2 * 16 = 2 * 4 * 4 = 2 * 2 * 2 * 2 * 2 = 25, so Quantity A = 5.
60 = 2 * 30 = 2 * 6 * 5 = 2 * 2 * 3 * 5 = 22 * 3 * 5, so Quantity B = 2.
Quantity A is greater. Even though 60 is a larger number than 32, 32 has more 2's in its prime factorization.
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If 
is an integer and 
is an integer, which of the following could be the value of 
?
If
Because 
, the answer choice that has a factorization set that cancels out completely with 396 will produce an integer. Only 18 fits this qualification, since 
.
Because
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What is the sum of the individual factors of 100 and 200?
What is the sum of the individual factors of 100 and 200?
Do not try to count out the factors. A neat formula for finding the sum of factors of a number can be utilized by first determining the prime factorization of the number.
,
where s is the sum, a, b, and c are factors, and x, y, and z are the powers of these factors.
Then, a = 2, b = 5, x = 2, y = 2.
Then, a = 2, b = 5, x = 3, y = 2.
Now we can add our two sums.
Do not try to count out the factors. A neat formula for finding the sum of factors of a number can be utilized by first determining the prime factorization of the number.
where s is the sum, a, b, and c are factors, and x, y, and z are the powers of these factors.
Then, a = 2, b = 5, x = 2, y = 2.
Then, a = 2, b = 5, x = 3, y = 2.
Now we can add our two sums.
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If the product of two distinct integer is 
, which of the following could not represent the sum of those two integers?
If the product of two distinct integer is
Since we're dealing with a product that comes out to a positive value, it could be the product of two positives or two negatives.
That being said, consider the ways we could factor 
:
For each of these four possible factors, there are four possible sums:
Since we're dealing with a product that comes out to a positive value, it could be the product of two positives or two negatives.
That being said, consider the ways we could factor
For each of these four possible factors, there are four possible sums:
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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
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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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
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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What is the largest possible integer value of if divides 16! evenly?
What is the largest possible integer value of
This question is really asking, “How many factors of 4 are there in 16!”? To ascertain this, list all the even numbers and count the total number of 2s among those factors.
Respectively, 16, 14, 12, 10, 8, 6, 4, 2 have 4, 1, 2, 1, 3, 1, 2, 1 factors of 2.
The total then is 15. This means that you have a factor of 215, which is the same as 47 * 2; therefore, since you are asked for the largest integer value of n, 7 is your answer.
Any larger integer value would not allow 4n to divide 16! evenly.
This question is really asking, “How many factors of 4 are there in 16!”? To ascertain this, list all the even numbers and count the total number of 2s among those factors.
Respectively, 16, 14, 12, 10, 8, 6, 4, 2 have 4, 1, 2, 1, 3, 1, 2, 1 factors of 2.
The total then is 15. This means that you have a factor of 215, which is the same as 47 * 2; therefore, since you are asked for the largest integer value of n, 7 is your answer.
Any larger integer value would not allow 4n to divide 16! evenly.
Compare your answer with the correct one above
Column A
5!/3!
Column B
6!/4!
Column A
5!/3!
Column B
6!/4!
This is a basic factorial question. A factorial is equal to the number times every positive whole number smaller than itself. In Column A, the numerator is 5 * 4 * 3 * 2 * 1 while the denominator is 3 * 2 * 1.
As you can see, the 3 * 2 * 1 can be cancelled out from both the numerator and denominator, leaving only 5 * 4.
The value for Column A is 5 * 4 = 20.
In Column B, the numerator is 6 * 5 * 4 * 3 * 2 * 1 while the denominator is 4 * 3 * 2 * 1. After simplifying, Column B gives a value of 6 * 5, or 30.
Thus, Column B is greater than Column A.
This is a basic factorial question. A factorial is equal to the number times every positive whole number smaller than itself. In Column A, the numerator is 5 * 4 * 3 * 2 * 1 while the denominator is 3 * 2 * 1.
As you can see, the 3 * 2 * 1 can be cancelled out from both the numerator and denominator, leaving only 5 * 4.
The value for Column A is 5 * 4 = 20.
In Column B, the numerator is 6 * 5 * 4 * 3 * 2 * 1 while the denominator is 4 * 3 * 2 * 1. After simplifying, Column B gives a value of 6 * 5, or 30.
Thus, Column B is greater than Column A.
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The prime factorization of 60 is?
The prime factorization of 60 is?
Prime numbers are numbers that can only divided by one and themselves. Breaking 60 into its prime factors yields:
Prime numbers are numbers that can only divided by one and themselves. Breaking 60 into its prime factors yields:
Compare your answer with the correct one above
Which of the following integers are factors of both 24 and 42?
Which of the following integers are factors of both 24 and 42?
3 is the only answer that is a factor of both 24 and 42. 42/3 = 14 and 24/3 = 8. The other answers are either a factor of 24 OR 42 or neither, but not both.
3 is the only answer that is a factor of both 24 and 42. 42/3 = 14 and 24/3 = 8. The other answers are either a factor of 24 OR 42 or neither, but not both.
Compare your answer with the correct one above
721(413) + 211(721) is equal to which of the following?
721(413) + 211(721) is equal to which of the following?
The answer is 721(413 + 211) because we can pull out a common factor, or 721, from both sides of the equation.
The answer is 721(413 + 211) because we can pull out a common factor, or 721, from both sides of the equation.
Compare your answer with the correct one above
n is a positive integer . p = 4 * 6 * 11 * n
Quantity A: The remainder when p is divided by 5
Quantity B: The remainder when p is divided by 33
n is a positive integer . p = 4 * 6 * 11 * n
Quantity A: The remainder when p is divided by 5
Quantity B: The remainder when p is divided by 33
Let's consider Quantity B first. What will the remainder be when p is divided by 33?
4, 6 and 11 are factors of p which means that 2 * 2 * 2 * 3 * 11 * n will equal p. We can group the 3 and 11 to see that 33 will always be a factor of p and will have no remainder. Thus Quantity B will always equal 0 no matter the value of n.
Now consider Quantity A. Let's consider first the values for p when n equals 1 through 5. When n = 1, p = 264, and the remainder is 4/5 or 0.8.
n = 2, p = 528, and the remainder is 3/5 or 0.6.
n = 3, p = 792, and the remainder is 2/5 or 0.4.
n = 4, p = 1056, and the remainder is 1/5 or 0.2.
n = 5, p = 1320, and the remainder is 0 (because when n = 5, 5 becomes a factor of p and thus there is no remainder.
Because Quantity A can be equal to or greater than B, there is not enough information given to determine the relationship.
Let's consider Quantity B first. What will the remainder be when p is divided by 33?
4, 6 and 11 are factors of p which means that 2 * 2 * 2 * 3 * 11 * n will equal p. We can group the 3 and 11 to see that 33 will always be a factor of p and will have no remainder. Thus Quantity B will always equal 0 no matter the value of n.
Now consider Quantity A. Let's consider first the values for p when n equals 1 through 5. When n = 1, p = 264, and the remainder is 4/5 or 0.8.
n = 2, p = 528, and the remainder is 3/5 or 0.6.
n = 3, p = 792, and the remainder is 2/5 or 0.4.
n = 4, p = 1056, and the remainder is 1/5 or 0.2.
n = 5, p = 1320, and the remainder is 0 (because when n = 5, 5 becomes a factor of p and thus there is no remainder.
Because Quantity A can be equal to or greater than B, there is not enough information given to determine the relationship.
Compare your answer with the correct one above
Quantitative Comparison
Quantity A: number of 2's in the prime factorization of 32
Quantity B: number of 2's in the prime factorization of 60
Quantitative Comparison
Quantity A: number of 2's in the prime factorization of 32
Quantity B: number of 2's in the prime factorization of 60
32 = 2 * 16 = 2 * 4 * 4 = 2 * 2 * 2 * 2 * 2 = 25, so Quantity A = 5.
60 = 2 * 30 = 2 * 6 * 5 = 2 * 2 * 3 * 5 = 22 * 3 * 5, so Quantity B = 2.
Quantity A is greater. Even though 60 is a larger number than 32, 32 has more 2's in its prime factorization.
32 = 2 * 16 = 2 * 4 * 4 = 2 * 2 * 2 * 2 * 2 = 25, so Quantity A = 5.
60 = 2 * 30 = 2 * 6 * 5 = 2 * 2 * 3 * 5 = 22 * 3 * 5, so Quantity B = 2.
Quantity A is greater. Even though 60 is a larger number than 32, 32 has more 2's in its prime factorization.
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If is an integer and is an integer, which of the following could be the value of ?
If
Because , the answer choice that has a factorization set that cancels out completely with 396 will produce an integer. Only 18 fits this qualification, since .
Because
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