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ISEE Upper Level Quantitative : Factors / Multiples

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

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Example Question #22 : How To Factor A Number

If we consider the factors of 1369 as a set of numbers, which one is greater?

(a) Product of the the median and the mean of the set

(b) The range of the set

Possible Answers:

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

(b) is greater

(a) is greater

(a) and (b) are equal

Correct answer:

(a) is greater

Explanation:

Factors of 1369 are $1,37\ and\ 1369$ . So we have:

$\left \{ 1,37,1369\right \}$

The range is the difference between the lowest and the highest values. So we have:

Range=1369-1=1368

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.

$Mean=\frac{1+37+1369}{3}=469$

The median is the middle value of a set of data containing an odd number of values:

Median=13

So we have:

$Mean\times Median=469\times 13=6097$

So (b) is greater than (a)

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Example Question #23 : How To Factor A Number

If we consider the factors of as a set of numbers, which one is greater?

(a) The mean of the set

(b) Ratio of the range and the median of the set

Possible Answers:

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

(a) and (b) are equal

(b) is greater

(a) is greater

Correct answer:

(a) is greater

Explanation:

Factors of are $1,5\ and\25$ . So we have:

$\left \{ 1,5,25\right \}$

The range is the difference between the lowest and the highest values. So we have:

Range=25-1=24

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:

$\frac{Range}{Median}=\frac{24}{5}=4.8$

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.

$Mean=\frac{1+5+25}{3}\approx 10.33$

So (a) is greater than (b)

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Example Question #24 : How To Factor A Number

Which one is greater?

(a) The total number of factors of

$(b)\ 10$

Possible Answers:

(a) is greater

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

(b) is greater

(a) and (b) are equal

Correct answer:

(b) is greater

Explanation:

Factors of are:

$1,2,3,4,6,9,12, 18\ and\ 36$ . So it has factors, which is less than .

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Example Question #25 : How To Factor A Number

Which one is the greater quantity:

(a) Sum of the factors of 125

(b) Sum of the factors of

Possible Answers:

(b) is greater

(a) is greater

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

(a) and (b) are equal

Correct answer:

(b) is greater

Explanation:

Factors of 125 are:

$1,5,25\ and\ 125$ and their summation is:

1+5+25+125=156

and the factors of are:

$1,2,4,5,8,10,16,20,40\ and\ 80$ and their summation is:

1+2+4+5+8+10+16+20+40+80=186

So (b) is greater than (a) .

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Example Question #26 : How To Factor A Number

If we consider the factors of 100 as a set of numbers, compare the mean and the median of the set.

Possible Answers:

The mean is greater

The median is greater

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

The mean and the median are equal

Correct answer:

The mean is greater

Explanation:

Factors of 100 are $1,2,4,5,10,20,25,50\ and\ 100$ . So we should compare the mean and the median of the following set of numbers:

$\left \{ 1,2,4,5,10,20,25,50,100 \right \}$

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:

$Mean=\frac{1+2+4+5+10+20+25+50+100}{9}=\frac{217}{9}\approx24$

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.

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Example Question #27 : How To Factor A Number

Which one is greater?

(a) The product of the factors of .

(b) The median of the following set:

$\left \{ 40,42,48,54,70 \right \}$

Possible Answers:

(a) and (b) are equal

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

(a) is greater

(b) is greater

Correct answer:

(a) is greater

Explanation:

Factors of are: $1,2,4\ and \8$ . So the product of the factors of are:

$1\times 2\times 4\times 8=64$

The median is the middle value of a set of data containing an odd number of values, which is in this problem. So (a) is greater than (b) .

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Example Question #28 : How To Factor A Number

Which one is greater?

(a) The sum of the factors of

(b) The product of the factors of

Possible Answers:

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

(a) and (b) are equal

(a) is greater

(b) is greater

Correct answer:

(b) is greater

Explanation:

Factors of are $1,2,3,4,5,6,10,12,15,20,30\ and\ 60$

$\Rightarrow Sum \ of\ the \ factors=1+2+3+4+5+6+10+12+15+20+30+60=168$

Factors of are $1,2,7\ and\14$

$\Rightarrow Product \ of\ the\ factors=1\times 2\times 7\times 14=196$

So (b) is greater thaan (a) .

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Example Question #29 : How To Factor A Number

and are positive integers; x > y ; x+ y is an even number. Which of the following also must be even?

Possible Answers:

x - y + 1

x - y

x + 2 y

xy + 1

Correct answer:

x - y

Explanation:

If x+ y is even, then and are either both even or both odd. The difference of two even numbers is even, and so is the difference of two odd numbers, so x - y must be even. Let's check the other choices, however:

x - y must be even, so x - y + 1 must be odd.

is either the product of two even numbers, which is even, or the product of two odd numbers, which is odd. Therefore, is of indeterminate sign. Similarly, xy + 1 is as well.

is even for any integer , so x + 2 y takes the same sign as ; but this is not given to us.

x - y is the correct choice.

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Example Question #21 : Other Factors / Multiples

Which is the greater quantity?

(A)

(B) The sum of the factors of 28 except for 28 itself.

Possible Answers:

(A) is greater

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

(A) and (B) are equal

(B) is greater

Correct answer:

(A) and (B) are equal

Explanation:

Leaving out 28 itself, the factors of 28 are $\left \{ 1, 2, 4, 7, 14\right \}$ . The sum of all of these factors is 1 + 2 + 4 + 7 + 14 = 28 , making the quantities equal.

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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)

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

Explanation:

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.

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Factors / Multiples

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #22 : How To Factor A Number

Example Question #23 : How To Factor A Number

Example Question #24 : How To Factor A Number

Example Question #25 : How To Factor A Number

Example Question #26 : How To Factor A Number

Example Question #27 : How To Factor A Number

Example Question #28 : How To Factor A Number

Example Question #29 : How To Factor A Number

Example Question #21 : Other Factors / Multiples

Example Question #31 : Numbers And Operations

All ISEE Upper Level Quantitative Resources

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