How to factor a number - ISEE Upper Level Quantitative Reasoning
Card 1 of 240
Which one is greater?
The product of the factors of 
.
The median of the following set:
Which one is greater?
Tap to reveal answer
Factors of 
are: 
. So the product of the factors of 
are:
The median is the middle value of a set of data containing an odd number of values, which is 
in this problem. So 
is greater than 
.
Factors of
The median is the middle value of a set of data containing an odd number of values, which is
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Which one is greater?
The product of the factors of .
The median of the following set:
Which one is greater?
Tap to reveal answer
Factors of are: . So the product of the factors of are:
The median is the middle value of a set of data containing an odd number of values, which is in this problem. So is greater than .
Factors of
The median is the middle value of a set of data containing an odd number of values, which is
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How many integers from 51 to 70 inclusive do not have 2, 3, or 5 as a factor?
How many integers from 51 to 70 inclusive do not have 2, 3, or 5 as a factor?
Tap to reveal answer
We can eliminate the ten even integers right off the bat, since, by definition, all have 
as a factor. Of the remaining (odd) integers, we eliminate 
and 
, as they have 
as a factor. What remains is:
We can now eliminate the multiples of 
. This leaves
.
The correct choice is 
.
We can eliminate the ten even integers right off the bat, since, by definition, all have
We can now eliminate the multiples of
The correct choice is
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How many integers from 51 to 70 inclusive do not have 2, 3, or 5 as a factor?
How many integers from 51 to 70 inclusive do not have 2, 3, or 5 as a factor?
Tap to reveal answer
We can eliminate the ten even integers right off the bat, since, by definition, all have as a factor. Of the remaining (odd) integers, we eliminate and , as they have as a factor. What remains is:
We can now eliminate the multiples of . This leaves
.
The correct choice is .
We can eliminate the ten even integers right off the bat, since, by definition, all have
We can now eliminate the multiples of
The correct choice is
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How many integers from 61 to 100 inclusive do not have 2, 3, 5, or 7 as a factor?
How many integers from 61 to 100 inclusive do not have 2, 3, 5, or 7 as a factor?
Tap to reveal answer
An integer is a multiple of 2 if and only if it ends in 2, 4, 6, 8, or 0; it is a multiple of 5 if and only if it ends in a 0 or 5. We can immediately eliminate these integers, leaving us with this set:
We eliminate the multiples of 3, which are 63, 69, 81, 87, 93, and 99:
We then eliminate the multiples of 7, which are 77 and 91:
.
This leaves eight elements.
An integer is a multiple of 2 if and only if it ends in 2, 4, 6, 8, or 0; it is a multiple of 5 if and only if it ends in a 0 or 5. We can immediately eliminate these integers, leaving us with this set:
We eliminate the multiples of 3, which are 63, 69, 81, 87, 93, and 99:
We then eliminate the multiples of 7, which are 77 and 91:
This leaves eight elements.
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Which one is greater?
The product of the factors of .
The median of the following set:
Which one is greater?
Tap to reveal answer
Factors of are: . So the product of the factors of are:
The median is the middle value of a set of data containing an odd number of values, which is in this problem. So is greater than .
Factors of
The median is the middle value of a set of data containing an odd number of values, which is
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Which one is greater?
The product of the factors of .
The median of the following set:
Which one is greater?
Tap to reveal answer
Factors of are: . So the product of the factors of are:
The median is the middle value of a set of data containing an odd number of values, which is in this problem. So is greater than .
Factors of
The median is the middle value of a set of data containing an odd number of values, which is
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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
Tap to reveal answer
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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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
Tap to reveal answer
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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If we consider the factors of 
as a set of numbers, compare the median and the range of the set.
If we consider the factors of
Tap to reveal answer
Factors of 
are 
. So we should compare the median and the range of the following set of numbers:
The range is the difference between the lowest and the highest values. So we have:
The median is the average of the two middle values of a set of data with an even number of values:
So the range is greater than the median.
Factors of
The range is the difference between the lowest and the highest values. So we have:
The median is the average of the two middle values of a set of data with an even number of values:
So the range is greater than the median.
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If we consider the factors of 
as a set of numbers, compare the mean and the range of the set.
If we consider the factors of
Tap to reveal answer
Factors of 
are 
. So we should compare the median and the range of the following set of numbers:
The range is the difference between the lowest and the highest values. So we have:
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.
So the range is greater than the mean.
Factors of
The range is the difference between the lowest and the highest values. So we have:
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.
So the range is greater than the mean.
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If we consider the factors of 
as a set of numbers, which one is greater?
The range of the set
Sum of the median and the mean of the set
If we consider the factors of
Tap to reveal answer
Factors of 
are 
. So we have:
The range is the difference between the lowest and the highest values. So we have:
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 average of the two middle values of a set of data with an even number of values:
So we have:
So 
is greater than 
Factors of
The range is the difference between the lowest and the highest values. So we have:
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 average of the two middle values of a set of data with an even number of values:
So we have:
So
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If we consider the factors of 
as a set of numbers, which one is greater?
Product of the the median and the mean of the set
The range of the set
If we consider the factors of
Tap to reveal answer
Factors of 
are 
. So we have:
The range is the difference between the lowest and the highest values. So we have:
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:
So we have:
So 
is greater than 
Factors of
The range is the difference between the lowest and the highest values. So we have:
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:
So we have:
So
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If we consider the factors of 
as a set of numbers, which one is greater?
The mean of the set
Ratio of the range and the median of the set
If we consider the factors of
Tap to reveal answer
Factors of 
are 
. So we have:
The range is the difference between the lowest and the highest values. So we have:
The median is the middle value of a set of data containing an odd number of values, which is 
in this problem. So the ratio of the range and the median is:
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.
So 
is greater than 
Factors of
The range is the difference between the lowest and the highest values. So we have:
The median is the middle value of a set of data containing an odd number of values, which is
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.
So
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Which is the greater quantity?
(A) 
(B) The sum of the factors of 28 except for 28 itself.
Which is the greater quantity?
(A)
(B) The sum of the factors of 28 except for 28 itself.
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Leaving out 28 itself, the factors of 28 are 
. The sum of all of these factors is 
, making the quantities equal.
Leaving out 28 itself, the factors of 28 are
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Which is the greater quantity?
(A) 60
(B) The sum of the factors of 60 except for 60 itself.
Which is the greater quantity?
(A) 60
(B) The sum of the factors of 60 except for 60 itself.
Tap to reveal answer
Leaving out 60 itself, the factors of 60 are 
. The sum of all of these factors can easily be seen to exceed 60, since the sum of the three largest factors is
.
This makes (B) greater.
Leaving out 60 itself, the factors of 60 are
This makes (B) greater.
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How many integers from 121 to 150 inclusive do not have 2, 3, or 5 as a factor?
How many integers from 121 to 150 inclusive do not have 2, 3, or 5 as a factor?
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An integer is a multiple of 2 if and only if it ends in 2, 4, 6, 8, or 0; it is a multiple of 5 if and only if it ends in a 0 or 5. We can immediately eliminate these integers, leaving us with this set:
Of these integers, the multiples of 3 are 123, 129, 141, and 147, leaving the set:
The correct response is eight.
An integer is a multiple of 2 if and only if it ends in 2, 4, 6, 8, or 0; it is a multiple of 5 if and only if it ends in a 0 or 5. We can immediately eliminate these integers, leaving us with this set:
Of these integers, the multiples of 3 are 123, 129, 141, and 147, leaving the set:
The correct response is eight.
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Which of the following is the greater quantity?
(A) The number of integers between 101 and 130 inclusive that do not have 2, 3, or 5 as a factor
(B) The number of integers between 201 and 230 inclusive that do not have 2, 3, or 5 as a factor
Which of the following is the greater quantity?
(A) The number of integers between 101 and 130 inclusive that do not have 2, 3, or 5 as a factor
(B) The number of integers between 201 and 230 inclusive that do not have 2, 3, or 5 as a factor
Tap to reveal answer
An integer is a multiple of 2 if and only if it ends in 2, 4, 6, 8, or 0; it is a multiple of 5 if and only if it ends in a 0 or 5. We can immediately eliminate these integers in each set.
In the set given in (A), we are left with
Eliminating the remaining multiples of 3, which are 111, 117, 123, and 129, we are left with
,
a set with eight elements.
Similarly, in the set given in (B), we are left with
.
Eliminating the remaining multiples of 3, which are 201, 207, 213, and 219, we are left with
,
a set with eight elements.
The quantities are equal.
An integer is a multiple of 2 if and only if it ends in 2, 4, 6, 8, or 0; it is a multiple of 5 if and only if it ends in a 0 or 5. We can immediately eliminate these integers in each set.
In the set given in (A), we are left with
Eliminating the remaining multiples of 3, which are 111, 117, 123, and 129, we are left with
a set with eight elements.
Similarly, in the set given in (B), we are left with
Eliminating the remaining multiples of 3, which are 201, 207, 213, and 219, we are left with
a set with eight elements.
The quantities are equal.
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How many integers from 61 to 100 inclusive do not have 2, 3, 5, or 7 as a factor?
How many integers from 61 to 100 inclusive do not have 2, 3, 5, or 7 as a factor?
Tap to reveal answer
An integer is a multiple of 2 if and only if it ends in 2, 4, 6, 8, or 0; it is a multiple of 5 if and only if it ends in a 0 or 5. We can immediately eliminate these integers, leaving us with this set:
We eliminate the multiples of 3, which are 63, 69, 81, 87, 93, and 99:
We then eliminate the multiples of 7, which are 77 and 91:
.
This leaves eight elements.
An integer is a multiple of 2 if and only if it ends in 2, 4, 6, 8, or 0; it is a multiple of 5 if and only if it ends in a 0 or 5. We can immediately eliminate these integers, leaving us with this set:
We eliminate the multiples of 3, which are 63, 69, 81, 87, 93, and 99:
We then eliminate the multiples of 7, which are 77 and 91:
This leaves eight elements.
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Which of the following is the greater quantity?
(A) The number of integers between 131 and 160 inclusive that do not have 2, 3, 5, or 7 as a factor
(B) The number of integers between 231 and 260 inclusive that do not have 2, 3, 5, or 7 as a factor
Which of the following is the greater quantity?
(A) The number of integers between 131 and 160 inclusive that do not have 2, 3, 5, or 7 as a factor
(B) The number of integers between 231 and 260 inclusive that do not have 2, 3, 5, or 7 as a factor
Tap to reveal answer
An integer is a multiple of 2 if and only if it ends in 2, 4, 6, 8, or 0; it is a multiple of 5 if and only if it ends in a 0 or 5. We can immediately eliminate these integers in each set.
In the set given in (A), we are left with
Eliminating the remaining multiples of 3, which are 141, 147, 153, and 159, we are left with
Of the remaining numbers, 133 is the only multiple of 7; we remove it, leaving the set
This leaves a set with seven elements.
In the set given in (B), we are left with
Eliminating the remaining multiples of 3, which are 231, 237, 243, and 249, we are left with
Of the remaining numbers, 259 is the only multiple of 7; we remove it, leaving the set
This leaves a set with seven elements.
The quantities are equal.
An integer is a multiple of 2 if and only if it ends in 2, 4, 6, 8, or 0; it is a multiple of 5 if and only if it ends in a 0 or 5. We can immediately eliminate these integers in each set.
In the set given in (A), we are left with
Eliminating the remaining multiples of 3, which are 141, 147, 153, and 159, we are left with
Of the remaining numbers, 133 is the only multiple of 7; we remove it, leaving the set
This leaves a set with seven elements.
In the set given in (B), we are left with
Eliminating the remaining multiples of 3, which are 231, 237, 243, and 249, we are left with
Of the remaining numbers, 259 is the only multiple of 7; we remove it, leaving the set
This leaves a set with seven elements.
The quantities are equal.
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