Other Factors / Multiples

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ISEE Upper Level Quantitative Reasoning › Other Factors / Multiples

Questions 1 - 10

Which is the greater quantity?

(a) The number of factors of 169

(b) The number of factors of 121

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Explanation

Each number has only three factors. 121 has 1, 11, and 121 as factors; 169 has 1, 13, and 169 as factors. The answer is that the quantities are equal.

Which is the greater quantity?

(a) The number of factors of 169

(b) The number of factors of 121

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Explanation

Each number has only three factors. 121 has 1, 11, and 121 as factors; 169 has 1, 13, and 169 as factors. The answer is that the quantities are equal.

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

The mean of the set

Ratio of the range and the median of the set

is greater

and are equal

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

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:

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 is greater than

Let be the set of all integers such that is divisible by and . How many elements are in ?

Explanation

The elements are as follows:

$A = \left { 5, 10, 15, 20...395 \right }$

This can be rewritten as

$A = \left { 5 \times 1, 5\times 2, ... 5 \times 79 \right }$ .

Therefore, there are elements in .

Which is the greater quantity?

(A)

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

(A) and (B) are equal

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

(A) is greater

(B) is greater

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 , making the quantities equal.

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

Product of the the median and the mean of the set

The range of the set

is greater

and are equal

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

Explanation

Factors of 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:

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:

So we have:

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

So is greater than

Which is the greater quantity?

(a) The number of factors of 15

(b) The number of factors of 17

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Explanation

(a) 15 has four factors, 1, 3, 5, and 15.

(b) 17, as a prime, has two factors, 1 and 17.

Therefore, (a) is greater.

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

The mean of the set

Ratio of the range and the median of the set

is greater

and are equal

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

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:

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 is greater than

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

The mean is greater

The median is greater

The mean and the median are equal

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

Explanation

Factors of 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.

Let be the set of all integers such that is divisible by and . How many elements are in ?

Explanation

The elements are as follows:

$A = \left { 5, 10, 15, 20...395 \right }$

This can be rewritten as

$A = \left { 5 \times 1, 5\times 2, ... 5 \times 79 \right }$ .

Therefore, there are elements in .

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