### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #31 : How To Factor A Number

is a positive even integer. If it is divided by 4, the remainder is . Which is the greater quantity?

(A)

(B)

**Possible Answers:**

(A) is greater

(B) is greater

It is impossible to tell which is greater from the information given

(A) and (B) are equal

**Correct answer:**

(A) is greater

An integer divided by 4 yields one of four remainders - 0, 1, 2, or 3. However, if the integer is even, the remainder must also be even, so it must be 0 or 2. Either way, 3 is greater, so (A) is the correct choice.

### Example Question #41 : Factors / Multiples

is a positive odd integer. If it is divided by 8, the remainder is 5. Which is the greater quantity?

(A) The remainder if is divided by 4

(B)

**Possible Answers:**

(A) is greater

(B) is greater

It is impossible to tell which is greater from the information given

(A) and (B) are equal

**Correct answer:**

(A) and (B) are equal

Since divided by 8 yields remainder 5, for some integer quotient ,

This can be rewritten as

Thus, divided by 4 yields quotient , which is an integer, and remainder 1. The quantities are equal.

### Example Question #32 : How To Factor A Number

Which is the greater quantity?

(A) 60

(B) The sum of the factors of 60 except for 60 itself.

**Possible Answers:**

(A) and (B) are equal

(B) is greater

(A) is greater

It is impossible to determine which is greater from the information given

**Correct answer:**

(B) is greater

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.

### Example Question #33 : How To Factor A Number

How many integers from 121 to 150 *inclusive* do not have 2, 3, or 5 as a factor?

**Possible Answers:**

Eight

Nine

Eleven

Seven

Ten

**Correct answer:**

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.

### Example Question #34 : How To Factor A Number

How many integers from 61 to 100 *inclusive* do not have 2, 3, 5, or 7 as a factor?

**Possible Answers:**

The correct answer is not given among the other responses.

Ten

Twelve

Nine

Eleven

**Correct answer:**

The correct answer is not given among the other responses.

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.

### Example Question #36 : Other Factors / Multiples

An integer is *abundant* if the sum of all of its factors, except for itself, is greater than . Of the following four integers, *how many* are abundant?

(A)

(B)

(C)

(D)

**Possible Answers:**

Two

One

None

Four

Three

**Correct answer:**

None

Add the factors of each number (except for the number itself) and compare to the number:

In each case, the sum of the factors is less than the number, so none of the integers given are abundant.

### Example Question #37 : Other Factors / Multiples

An integer is *deficient *if the sum of all of its factors, except for itself, is less than . Of the following four integers, *how many* are deficient?

(A)

(B)

(C)

(D)

**Possible Answers:**

Four

One

Two

None

Three

**Correct answer:**

Three

Add the factors of each number (except for the number itself) and compare to the number:

26, 46, and 86 all have factor sums less than themselves, so the correct response is "three".

### Example Question #32 : How To Factor A Number

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

**Possible Answers:**

It is impossible to determine which is greater from the information given

(B) is greater

(A) is greater

(A) and (B) are equal

**Correct answer:**

(A) and (B) 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.

### Example Question #45 : Factors / Multiples

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

**Possible Answers:**

(A) and (B) are equal

(B) is greater

It is impossible to determine which is greater from the information given

(A) is greater

**Correct answer:**

(A) and (B) 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.

### Example Question #35 : How To Factor A Number

Which of the following is true about the number 125?

**Possible Answers:**

It is larger than .

It is the square of a real number.

It is a multiple of 25.

It is a multiple of 75.

**Correct answer:**

It is a multiple of 25.

Given that , it follows that 25 is a multiple of 125; therefore, the answer choice, "It is a multiple of 25" is the correct answer.

125 does not have a real-number square root, so it is not the square of a real number (real numbers are those numbers, both rational and irrational, found on a number line). It is also not divisible by 75, and it is smaller that , which is equal to .