### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #11 : How To Factor A Number

Which is the greater quantity?

(a) The sum of all of the two-digit even numbers

(b) 2,500

**Possible Answers:**

(a) is greater

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

**Correct answer:**

(b) is greater

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:

### Example Question #21 : Numbers And Operations

What is the prime factorization of ?

**Possible Answers:**

**Correct answer:**

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 .

### Example Question #22 : Numbers And Operations

Which one is greater?

number of factors of

**Possible Answers:**

it is not possible to tell based on the information given.

is greater

is greater

and are equal

**Correct answer:**

is greater

Factors of are:

. So it has factors which is less than .

### Example Question #23 : Numbers And Operations

Which one is greater?

The sum of all of the factors of

**Possible Answers:**

and are equal

is greater

it is not possible to tell based on the information given.

is greater

**Correct answer:**

is greater

Factors of are:

. So we can write:

### Example Question #24 : Numbers And Operations

Which one is greater?

The number of factors of

The number of factors of

**Possible Answers:**

It is not possible to tell based on the information given.

is greater

and are equal

is greater

**Correct answer:**

and are equal

has only three factors of

and

also has three factors of

So the number of their factors are the same

### Example Question #25 : Numbers And Operations

Which one is greater?

The number of factors of

The number of factors of

**Possible Answers:**

It is not possible to tell based on the information given.

is greater

is greater

and are equal

**Correct answer:**

is greater

The factors of are . So has ten factors.

The factors of are . So has eight factors.

So is greater than .

### Example Question #12 : How To Factor A Number

If we consider the factors of as a set of numbers, compare the mean and the median of the set.

**Possible Answers:**

The mean and the median are equal

It is not possible to tell based on the information given.

The median is greater

The mean is greater

**Correct answer:**

The mean is greater

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.

### Example Question #27 : Numbers And Operations

If we consider the factors of as a set of numbers, compare the mean and the range of the set.

**Possible Answers:**

The range is greater

The mean is greater

The mean and the range are equal

It is not possible to tell based on the information given.

**Correct answer:**

The range is greater

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.

### Example Question #28 : Numbers And Operations

If we consider the factors of as a set of numbers, compare the median and the range of the set.

**Possible Answers:**

It is not possible to tell based on the information given.

The range and the median are equal

The median is greater

The range is greater

**Correct answer:**

The range is greater

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.

### Example Question #29 : Numbers And Operations

Which one is greater?

The sum of the factors of

The sum of the factors of

**Possible Answers:**

and are equal

It is not possible to tell based on the information given.

is greater

is greater

**Correct answer:**

is greater

Factors of are:

Factors of are:

So is greater than