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ISEE Upper Level Quantitative : Numbers and Operations

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

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #11 : Other Factors / Multiples

What is the prime factorization of 16ab^2 $\displaystyle 16ab^2$ ?

Possible Answers:

$2\cdot 2\cdot 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$ $\displaystyle 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$

$2\cdot 2\cdot 2\cdot a\cdot b\cdot b$ $\displaystyle 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$

$2\cdot 2\cdot 2\cdot 2\cdot a\cdot b$ $\displaystyle 2\cdot 2\cdot 2\cdot 2\cdot a\cdot b$

$2\cdot 2\cdot 2\cdot 2$ $\displaystyle 2\cdot 2\cdot 2\cdot 2$

$2\cdot 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$ $\displaystyle 2\cdot 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$

Correct answer:

$2\cdot 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$ $\displaystyle 2\cdot 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$

Explanation:

First make a factor tree for 16. Keep breaking it down until you get all prime numbers (for example: $4\times 4$ $\displaystyle 4\times 4$ , which then yields $2\times 2\times 2\times 2$ $\displaystyle 2\times 2\times 2\times 2$ ). 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 $2\cdot 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$ $\displaystyle 2\cdot 2\cdot 2\cdot 2\cdot a\cdot b\cdot b$ .

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Example Question #12 : Other Factors / Multiples

Which one is greater?

(a) $\displaystyle (a)$ number of factors of 100 $\displaystyle 100$

$(b)\ 10$ $\displaystyle (b)\ 10$

Possible Answers:

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

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

(b) $\displaystyle (b)$ is greater

(a) $\displaystyle (a)$ is greater

Correct answer:

(b) $\displaystyle (b)$ is greater

Explanation:

Factors of 100 $\displaystyle 100$ are:

$1,2,4,5,10,20,25,50\ and\ 100$ $\displaystyle 1,2,4,5,10,20,25,50\ and\ 100$ . So it has $\displaystyle 9$ factors which is less than $\displaystyle 10$ .

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Example Question #13 : Other Factors / Multiples

Which one is greater?

(a) $\displaystyle (a)$ The sum of all of the factors of $\displaystyle 72$

$(b)\ 200$ $\displaystyle (b)\ 200$

Possible Answers:

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

(a) $\displaystyle (a)$ is greater

(b) $\displaystyle (b)$ is greater

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

Correct answer:

(b) $\displaystyle (b)$ is greater

Explanation:

Factors of $\displaystyle 72$ are:

$1,2,3,4,6,8,9,12,18,24,36\ and\ 72$ $\displaystyle 1,2,3,4,6,8,9,12,18,24,36\ and\ 72$ . So we can write:

$\displaystyle 1+2+3+4+6+8+9+12+18+24+36+72=195$

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Example Question #14 : Other Factors / Multiples

Which one is greater?

(a) $\displaystyle (a)$ The number of factors of 289 $\displaystyle 289$

(b) $\displaystyle (b)$ The number of factors of 361 $\displaystyle 361$

Possible Answers:

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

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

(a) $\displaystyle (a)$ is greater

(b) $\displaystyle (b)$ is greater

Correct answer:

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

Explanation:

289 $\displaystyle 289$ has only three factors of $1,17\ and\ 289$ $\displaystyle 1,17\ and\ 289$

and

361 $\displaystyle 361$ also has three factors of $1,19\ and\ 361$ $\displaystyle 1,19\ and\ 361$

So the number of their factors are the same

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Example Question #15 : Other Factors / Multiples

Which one is greater?

(a) $\displaystyle (a)$ The number of factors of $\displaystyle 80$

(b) $\displaystyle (b)$ The number of factors of $\displaystyle 70$

Possible Answers:

(b) $\displaystyle (b)$ is greater

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

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

(a) $\displaystyle (a)$ is greater

Correct answer:

(a) $\displaystyle (a)$ is greater

Explanation:

The factors of $\displaystyle 80$ are $1,2,4,5,8,10,16,20,40\ and\ 80$ $\displaystyle 1,2,4,5,8,10,16,20,40\ and\ 80$ . So $\displaystyle 80$ has ten factors.

The factors of $\displaystyle 70$ are $1,2,5,7,10,14,35\ and\ 70$ $\displaystyle 1,2,5,7,10,14,35\ and\ 70$ . So $\displaystyle 70$ has eight factors.

So (a) $\displaystyle (a)$ is greater than (b) $\displaystyle (b)$ .

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Example Question #16 : Other Factors / Multiples

If we consider the factors of 169 $\displaystyle 169$ as a set of numbers, compare the mean and the median of the set.

Possible Answers:

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

The mean is greater

The median is greater

The mean and the median are equal

Correct answer:

The mean is greater

Explanation:

Factors of 169 $\displaystyle 169$ are $1,13\ and\ 169$ $\displaystyle 1,13\ and\ 169$ . So we should compare the mean and the median of the following set of numbers:

$\left \{ 1,13,169\right \}$ $\displaystyle \left \{ 1,13,169\right \}$

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:

$Mean=\frac{1+13+169}{3}=61$ $\displaystyle Mean=\frac{1+13+169}{3}=61$

The median is the middle value of a set of data containing an odd number of values which is $\displaystyle 13$ in this problem. So the mean is greater than the median.

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Example Question #17 : Other Factors / Multiples

If we consider the factors of $\displaystyle 24$ as a set of numbers, compare the mean and the range of the set.

Possible Answers:

The mean and the range are equal

The mean is greater

The range is greater

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

Correct answer:

The range is greater

Explanation:

Factors of $\displaystyle 24$ are $1,2,3,4,6,8,12\ and\ 24$ $\displaystyle 1,2,3,4,6,8,12\ and\ 24$ . So we should compare the median and the range of the following set of numbers:

$\left \{ 1,2,3,4,6,8,12,24\right \}$ $\displaystyle \left \{ 1,2,3,4,6,8,12,24\right \}$

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

$\displaystyle Range=24-1=23$

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.

$Mean=\frac{1+2+3+4+6+8+12+24}{8}=7.5$ $\displaystyle Mean=\frac{1+2+3+4+6+8+12+24}{8}=7.5$

So the range is greater than the mean.

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Example Question #18 : Other Factors / Multiples

If we consider the factors of $\displaystyle 72$ 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 is greater

The range and the median are equal

The median is greater

Correct answer:

The range is greater

Explanation:

Factors of $\displaystyle 72$ are $1,2,3,4,6,8,9,12,18,24,36\ and\ 72$ $\displaystyle 1,2,3,4,6,8,9,12,18,24,36\ and\ 72$ . So we should compare the median and the range of the following set of numbers:

$\left \{ 1,2,3,4,6,8,9,12,18,24,36,72\right \}$ $\displaystyle \left \{ 1,2,3,4,6,8,9,12,18,24,36,72\right \}$

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

$\displaystyle Range=72-1=71$

The median is the average of the two middle values of a set of data with an even number of values:

$Median=\frac{8+9}{2}=8.5$ $\displaystyle Median=\frac{8+9}{2}=8.5$

So the range is greater than the median.

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Example Question #19 : Other Factors / Multiples

Which one is greater?

(a) $\displaystyle (a)$ The sum of the factors of $\displaystyle 24$

(b) $\displaystyle (b)$ The sum of the factors of $\displaystyle 28$

Possible Answers:

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

(b) $\displaystyle (b)$ is greater

(a) $\displaystyle (a)$ is greater

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

Correct answer:

(a) $\displaystyle (a)$ is greater

Explanation:

Factors of $\displaystyle 24$ are: $1,2,3,4,6,8,12 \ and\ 24$ $\displaystyle 1,2,3,4,6,8,12 \ and\ 24$

$\Rightarrow Sum\ of\ the \ factors=1+2+3+4+6+8+12+24=60$ $\displaystyle \Rightarrow Sum\ of\ the \ factors=1+2+3+4+6+8+12+24=60$

Factors of $\displaystyle 28$ are: $1,2,4,7,14\ and\ 28$ $\displaystyle 1,2,4,7,14\ and\ 28$

$\Rightarrow Sum \ of\ the\ factors=1+2+4+7+14+28=56$ $\displaystyle \Rightarrow Sum \ of\ the\ factors=1+2+4+7+14+28=56$

So (a) $\displaystyle (a)$ is greater than (b) $\displaystyle (b)$

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Example Question #21 : Other Factors / Multiples

If we consider the factors of $\displaystyle 45$ as a set of numbers, which one is greater?

(a) $\displaystyle (a)$ The range of the set

(b) $\displaystyle (b)$ Sum of the median and the mean of the set

Possible Answers:

(b) $\displaystyle (b)$ is greater

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

(a) $\displaystyle (a)$ is greater

(a) $\displaystyle (a)$ and (b) $\displaystyle (b)$ are equal

Correct answer:

(a) $\displaystyle (a)$ is greater

Explanation:

Factors of $\displaystyle 45$ are $1,3,5,9,15\ and\ 45$ $\displaystyle 1,3,5,9,15\ and\ 45$ . So we have:

$\left \{ 1,3,5,9,15,45\right \}$ $\displaystyle \left \{ 1,3,5,9,15,45\right \}$

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

$\displaystyle Range=45-1=44$

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.

$Mean=\frac{1+3+5+9+15+45}{6}=13$ $\displaystyle Mean=\frac{1+3+5+9+15+45}{6}=13$

The median is the average of the two middle values of a set of data with an even number of values:

$Median=\frac{5+9}{2}=7$ $\displaystyle Median=\frac{5+9}{2}=7$

So we have:

$\displaystyle Mean+Median=13+7=20$

So (a) $\displaystyle (a)$ is greater than (b) $\displaystyle (b)$

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

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ISEE Upper Level Quantitative : Numbers and Operations

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #11 : Other Factors / Multiples

Example Question #12 : Other Factors / Multiples

Example Question #13 : Other Factors / Multiples

Example Question #14 : Other Factors / Multiples

Example Question #15 : Other Factors / Multiples

Example Question #16 : Other Factors / Multiples

Example Question #17 : Other Factors / Multiples

Example Question #18 : Other Factors / Multiples

Example Question #19 : Other Factors / Multiples

Example Question #21 : Other Factors / Multiples

All ISEE Upper Level Quantitative Resources

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