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ISEE Upper Level Quantitative : How to factor a number

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 : 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:

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

Correct answer:

(b) is greater

Explanation:

The sum of the integers from to is equal to $\frac{\left ( n+m \right )\left ( n-m+1 \right )}{2}$ . 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:

10+12 + 14+...+98

$=2 \left ( 5+ 6+7+...+49\right )$

$=2\cdot \frac{\left ( 49+5 \right )\left ( 49-5+1 \right )}{2}$

$=54 \times 45 = 2,430 < 2,500$

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Example Question #21 : Numbers And Operations

What is the prime factorization of 16ab^2 ?

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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Example Question #22 : Numbers And Operations

Which one is greater?

(a) number of factors of 100

$(b)\ 10$

Possible Answers:

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

(a) is greater

(b) is greater

(a) and (b) are equal

Correct answer:

(b) is greater

Explanation:

Factors of 100 are:

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

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Example Question #23 : Numbers And Operations

Which one is greater?

(a) The sum of all of the factors of

$(b)\ 200$

Possible Answers:

(a) and (b) are equal

(a) is greater

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

(b) is greater

Correct answer:

(b) is greater

Explanation:

Factors of are:

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

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

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Example Question #24 : Numbers And Operations

Which one is greater?

(a) The number of factors of 289

(b) The number of factors of 361

Possible Answers:

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

(a) is greater

(a) and (b) are equal

(b) is greater

Correct answer:

(a) and (b) are equal

Explanation:

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

and

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

So the number of their factors are the same

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Example Question #25 : Numbers And Operations

Which one is greater?

(a) The number of factors of

(b) The number of factors of

Possible Answers:

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

(a) is greater

(b) is greater

(a) and (b) are equal

Correct answer:

(a) is greater

Explanation:

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

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

So (a) is greater than (b) .

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Example Question #12 : How To Factor A Number

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

Possible Answers:

The mean and the median are equal

The median is greater

The mean is greater

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

Correct answer:

The mean is greater

Explanation:

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

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

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:

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

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 : 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

Explanation:

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

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

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

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.

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

So the range is greater than the mean.

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Example Question #13 : How To Factor A Number

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

Possible Answers:

The range is greater

The median is greater

The range and the median are equal

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

Correct answer:

The range is greater

Explanation:

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

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

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

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:

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

So the range is greater than the median.

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Example Question #29 : Numbers And Operations

Which one is greater?

(a) The sum of the factors of

(b) The sum of the factors of

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: $1,2,3,4,6,8,12 \ and\ 24$

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

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

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

So (a) is greater than (b)

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : How to factor a number

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #11 : How To Factor A Number

Example Question #21 : Numbers And Operations

Example Question #22 : Numbers And Operations

Example Question #23 : Numbers And Operations

Example Question #24 : Numbers And Operations

Example Question #25 : Numbers And Operations

Example Question #12 : How To Factor A Number

Example Question #27 : Numbers And Operations

Example Question #13 : How To Factor A Number

Example Question #29 : Numbers And Operations

All ISEE Upper Level Quantitative Resources

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