All ISEE Upper Level Quantitative Resources
Example Questions
Example Question #31 : Numbers And Operations
is a positive even integer. If it is divided by 4, the remainder is . Which is the greater quantity?
(A)
(B)
(A) is greater
(A) and (B) are equal
It is impossible to tell which is greater from the information given
(B) is greater
(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 #31 : Other 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)
(A) and (B) are equal
(A) is greater
It is impossible to tell which is greater from the information given
(B) is greater
(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 : Other Factors / Multiples
Which is the greater quantity?
(A) 60
(B) The sum of the factors of 60 except for 60 itself.
(A) is greater
(A) and (B) are equal
It is impossible to determine which is greater from the information given
(B) is greater
(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 : Other Factors / Multiples
How many integers from 121 to 150 inclusive do not have 2, 3, or 5 as a factor?
Ten
Eight
Seven
Nine
Eleven
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 : Other Factors / Multiples
How many integers from 61 to 100 inclusive do not have 2, 3, 5, or 7 as a factor?
Twelve
Ten
The correct answer is not given among the other responses.
Nine
Eleven
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)
Two
One
None
Four
Three
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 #36 : How To Factor A Number
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)
One
Three
Four
Two
None
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 #35 : Other Factors / Multiples
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
It is impossible to determine which is greater from the information given
(A) and (B) are equal
(A) is greater
(B) is greater
(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 #36 : Other 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
(A) is greater
(A) and (B) are equal
It is impossible to determine which is greater from the information given
(B) is greater
(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 #37 : Other Factors / Multiples
Which of the following is true about the number 125?
It is the square of a real number.
It is a multiple of 25.
It is a multiple of 75.
It is larger than .
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 .