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Example Questions
Example Question #45 : Understanding Real Numbers
Define an operation  on the integers as follows:
If both  andÂ
 are odd, thenÂ
.
If both  andÂ
 are even, thenÂ
.
If  is odd andÂ
 is even, or vice versa, thenÂ
.
Add  toÂ
. What is the sum?
Both  andÂ
 can be calculated using the definition of
 for the case of exactly one ofÂ
 andÂ
 being odd and one being even:
.
Add:Â
Example Question #46 : Understanding Real Numbers
Define an operation  on the integers as follows:
If both  andÂ
 are prime, thenÂ
.
If neither  norÂ
 is prime, thenÂ
.
If exactly one of   andÂ
 is prime, thenÂ
.
Multiply  byÂ
. What is the product?
A prime number has exactly two factors, 1 and the number itself.
Neither 6 nor 1 is a prime number; 1 has only one factor and is not considered to be prime, and 6 has more than two factors - 1, 2, 3, and 6. Therefore,  can be evaluated using the defintion ofÂ
 for two numbers whose absolute values are not prime:
2 and 3 are prime numbers, since each has exactly two factors, 1 and the number itself. Therefore,  can be evaluated using the defintion ofÂ
 for two numbers whose absolute values are prime:
The product isÂ
Example Question #47 : Understanding Real Numbers
Define an operation  on the integers as follows:
If both  andÂ
 are prime, thenÂ
.
If neither  norÂ
 is prime, thenÂ
.
If exactly one of   andÂ
 is prime, thenÂ
.
Subtract  fromÂ
. What is the result?
2 is a prime number, since 2 has only two factors, 1 and 2 itself. 50 is not a prime number, since 50 has other factors, such as 2.  can be evaluated using the definition ofÂ
 for exactly one ofÂ
 andÂ
 prime:
Â
Neither 4 nor 25 are prime, since each has factors other than 1 and itself; for example,  andÂ
.Â
 can be evaluated using the definition ofÂ
 for neitherÂ
 norÂ
 prime:
The difference:
Example Question #42 : Understanding Real Numbers
 is the additive inverse ofÂ
. Which of the following expressions is equivalent toÂ
for all values of the variables?
If  is the additive inverse ofÂ
, thenÂ
, or, equivalently,
By way of substitution and the identity property of addition,
Example Question #49 : Understanding Real Numbers
Define an operation  on the integers as follows:
If both  andÂ
 are prime, thenÂ
.
If neither  norÂ
 is prime, thenÂ
.
If exactly one of   andÂ
 is prime, thenÂ
.
Evaluate .
17 and 13 are both prime numbers, since each has exactly two factors - 1 and the number itself. Therefore, we first evaluate  using the definition ofÂ
 forÂ
andÂ
both prime:
Therefore, . 7 is also prime, since its only two factors are 1 and 7 itself. 30, however, is not prime, since 30 has factors other than 1 and itself - for example,Â
. Therefore,Â
 is evaluated using the definition ofÂ
 for exactly one ofÂ
andÂ
prime:
, the correct response.
Example Question #50 : Understanding Real Numbers
Define an operation  on the integers as follows:
If both  andÂ
 are prime, then
.
If neither  norÂ
 is prime, then
.
If exactly one of   andÂ
 is prime, thenÂ
.
Which of the following expressions is the greatest of the five?
Of the integers shown in the five choices, the following are primes, since they have exactly two factors, 1 and the number itself:Â 2, 5.
1 is not consdered to be a prime, having exactly one factor (1). Also, 4, 10, 20, 25, 50, and 100 are not primes, since each has at least one factor other than 1 and itself.
 and
can both be evaluated using the definition of
for exactly one of
and
prime - that is, by multiplying the numbers:
Each of ,
, andÂ
can be evaluated using the definition of
for neither of
and
prime - that is, by adding the numbers:
The greatest of the five expressions is .
Â
Example Question #2001 : Problem Solving Questions
Consider the following set of numbers:
85, 87, 87, 82, 89
What is the range?
The range is the difference between the maximum and minimum value.
Example Question #2 : Descriptive Statistics
What is the range for the following data set:
The range is the highest value number minus the lowest value number in a sorted data set:
We need to sort the data set:
Example Question #1 : Calculating Range
What is the range for the following set:
The range is the difference between the highest and lowest number.
First sort the set:
Example Question #1 : Descriptive Statistics
Below is the stem-and-leaf display of a set of test scores.
What is the range of this set of scores?
The range of a data set is the difference of the highest and lowest scores,
The numbers in the "stem" of this display represent tens digits of the test scores, and the numbers in the "leaves" represent the units digits. The highest and lowest scores represented are 87 and 42, so the range is their difference:Â .
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