How to factor a number - ISEE Upper Level Quantitative Reasoning
Card 0 of 240
Which is the greater quantity?
(a) The number of factors of 169
(b) The number of factors of 121
Which is the greater quantity?
(a) The number of factors of 169
(b) The number of factors of 121
Each number has only three factors. 121 has 1, 11, and 121 as factors; 169 has 1, 13, and 169 as factors. The answer is that the quantities are equal.
Each number has only three factors. 121 has 1, 11, and 121 as factors; 169 has 1, 13, and 169 as factors. The answer is that the quantities are equal.
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Which of the following is NOT a factor of 
?
Which of the following is NOT a factor of
First, we must solve for 
While 64 is divisible by 4, 8, and 16, it is not divisible by 7; therefore, 7 is not a factor of 64 and is thus the correct answer.
First, we must solve for
While 64 is divisible by 4, 8, and 16, it is not divisible by 7; therefore, 7 is not a factor of 64 and is thus the correct answer.
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Give the prime factorization of 91.
Give the prime factorization of 91.
Both are prime factors so this is the prime factorization.
Both are prime factors so this is the prime factorization.
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Add all of the factors of 30.
Add all of the factors of 30.
The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. Their sum is
.
The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. Their sum is
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How many factors does 40 have?
How many factors does 40 have?
40 has as its factors 1, 2, 4, 5, 8, 10, 20, and 40 - a total of eight factors.
40 has as its factors 1, 2, 4, 5, 8, 10, 20, and 40 - a total of eight factors.
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Which of these numbers is relatively prime with 18?
Which of these numbers is relatively prime with 18?
For two numbers to be relatively prime, they cannot have any factor in common except for 1. The factors of 18 are 1, 2, 3, 6, 9, and 18.
We can eliminate 32 and 34, since each shares with 18 a factor of 2; we can also eliminate 33 and 39, since each shares with 18 a factor of 3. The factors of 35 are 1, 5, 7, and 35; as can be seen by comparing factors, 18 and 35 only have 1 as a factor, making 35 the correct choice.
For two numbers to be relatively prime, they cannot have any factor in common except for 1. The factors of 18 are 1, 2, 3, 6, 9, and 18.
We can eliminate 32 and 34, since each shares with 18 a factor of 2; we can also eliminate 33 and 39, since each shares with 18 a factor of 3. The factors of 35 are 1, 5, 7, and 35; as can be seen by comparing factors, 18 and 35 only have 1 as a factor, making 35 the correct choice.
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Which of the following is the prime factorization of 333?
Which of the following is the prime factorization of 333?
To find the prime factorization, break the number down as a product of factors, then keep doing this until all of the factors are prime.
To find the prime factorization, break the number down as a product of factors, then keep doing this until all of the factors are prime.
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What is the sum of all of the factors of 27?
What is the sum of all of the factors of 27?
27 has four factors: 
Their sum is 
.
27 has four factors:
Their sum is
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Which is the greater quantity?
(a) The number of factors of 15
(b) The number of factors of 17
Which is the greater quantity?
(a) The number of factors of 15
(b) The number of factors of 17
(a) 15 has four factors, 1, 3, 5, and 15.
(b) 17, as a prime, has two factors, 1 and 17.
Therefore, (a) is greater.
(a) 15 has four factors, 1, 3, 5, and 15.
(b) 17, as a prime, has two factors, 1 and 17.
Therefore, (a) is greater.
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Which is the greater quantity?
(a) The product of the integers between 
and 
inclusive
(b) The sum of the integers between 
and 
inclusive
Which is the greater quantity?
(a) The product of the integers between
(b) The sum of the integers between
The quanitites are equal, as both can be demonstrated to be equal to 
.
(a) One of the integers in the given range is , so one of the factors will be , making the product .
(b) The sum of the numbers will be:
The quanitites are equal, as both can be demonstrated to be equal to .
(a) One of the integers in the given range is , so one of the factors will be
, making the product
.
(b) The sum of the numbers will be:
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Which is the greater quantity?
(a) The sum of the factors of 
(b) The sum of the factors of 
Which is the greater quantity?
(a) The sum of the factors of
(b) The sum of the factors of
(a) The factors of 
are 
Their sum is
.
(b) The factors of 
are 
Their sum is
.
(b) is greater.
(a) The factors of
(b) The factors of
(b) is greater.
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Which is the greater quantity?
(a) The sum of all of the two-digit even numbers
(b) 2,500
Which is the greater quantity?
(a) The sum of all of the two-digit even numbers
(b) 2,500
The sum of the integers from 
to 
is equal to 
. We take advantage of the fact that the sum of the even numbers from 10 to 98 is equal to twice the sum of the integers from 5 to 49, as seen here:
The sum of the integers from
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What is the prime factorization of 
?
What is the prime factorization of
First make a factor tree for 16. Keep breaking it down until you get all prime numbers (for example: 
, which then yields 
). Then, at the end, remember to factor the variables as well. Since the b term is squared, that means there are two of them. Therefore, the final answer is 
.
First make a factor tree for 16. Keep breaking it down until you get all prime numbers (for example:
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Which one is greater?
The total number of factors of 
Which one is greater?
Factors of 
are:
. So it has 
factors, which is less than 
.
Factors of
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Which one is greater?
number of factors of 
Which one is greater?
Factors of 
are:
. So it has 
factors which is less than 
.
Factors of
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Which one is greater?
The sum of all of the factors of 
Which one is greater?
Factors of 
are:
. So we can write:
Factors of
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Which one is the greater quantity:
Sum of the factors of 
Sum of the factors of 
Which one is the greater quantity:
Factors of 
are:
and their summation is:
and the factors of 
are:
and their summation is:
So 
is greater than 
.
Factors of
and the factors of
So
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Which one is greater?
The number of factors of 
The number of factors of 
Which one is greater?
has only three factors of 
and
also has three factors of 
So the number of their factors are the same
and
So the number of their factors are the same
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Which one is greater?
The number of factors of 
The number of factors of 
Which one is greater?
The factors of 
are 
. So 
has ten factors.
The factors of 
are 
. So 
has eight factors.
So 
is greater than 
.
The factors of
The factors of
So
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If we consider the factors of 
as a set of numbers, compare the mean and the median of the set.
If we consider the factors of
Factors of 
are 
. So we should compare the mean and the median of the following set of numbers:
The mean of a set of data is given by the sum of the data, divided by the total number of values in the set:
The median is the middle value of a set of data containing an odd number of values which is 
in this problem. So the mean is greater than the median.
Factors of
The mean of a set of data is given by the sum of the data, divided by the total number of values in the set:
The median is the middle value of a set of data containing an odd number of values which is
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