Types of Numbers and Number Theory - GED Math
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Express 
as a number in base ten.
Express
Each digit of a base six number has a place value that is a power of six. The lowest four powers of 6, beginning with 1, are 
, so
Each digit of a base six number has a place value that is a power of six. The lowest four powers of 6, beginning with 1, are
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Which of the following base ten numbers has a base sixteen representation of exactly three digits?
Which of the following base ten numbers has a base sixteen representation of exactly three digits?
A number in base sixteen has powers of sixteen as its place values; starting at the right, they are 
.
The lowest base sixteen number with three digits would be
in base ten.
The lowest base sixteen number with four digits would be
in base ten.
Therefore, a number that is expressed as a three-digit number in base sixteen must fall in the range
.
Of the four numbers listed, 1,000 falls in that range.
A number in base sixteen has powers of sixteen as its place values; starting at the right, they are
The lowest base sixteen number with three digits would be
The lowest base sixteen number with four digits would be
Therefore, a number that is expressed as a three-digit number in base sixteen must fall in the range
Of the four numbers listed, 1,000 falls in that range.
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How many elements of the set 
are rational?
How many elements of the set
0 and 1 are both integers and are therefore rational numbers. 
is known to be an irrational number, since there are no two integers whose quotient is equal to 
. "Two" is correct.
0 and 1 are both integers and are therefore rational numbers.
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How many elements of the set 
are rational?
How many elements of the set
The square root of an integer is irrational if it is not an integer itself. As can be calculated, all three of these square roots are not integers, so none are rational.
The square root of an integer is irrational if it is not an integer itself. As can be calculated, all three of these square roots are not integers, so none are rational.
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How many elements are in a set that has exactly 64 subsets?
How many elements are in a set that has exactly 64 subsets?
A set with 
elements has 
subsets. Since 
, a set with 64 subsets has 6 elements.
A set with
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How many subsets does the set 
have?
How many subsets does the set
The number of subsets of a set with 
elements is 
, so this eight-element set has 
subsets.
The number of subsets of a set with
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To how many of the following sets does the number 
belong?
I) The set of whole numbers
II) The set of integers
III) The set of rational numbers
To how many of the following sets does the number
I) The set of whole numbers
II) The set of integers
III) The set of rational numbers
, which is not an integer. It is not a whole number either, as the whole numbers consist of all (nonnegative) integers.
. As the quotient of integers, it is rational.
Therefore, 
belongs to the set of rational numbers but not to the other two sets. The correct response is one.
Therefore,
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To how many of the following sets does the number 
belong?
I) The set of whole numbers
II) The set of integers
III) The set of rational numbers
To how many of the following sets does the number
I) The set of whole numbers
II) The set of integers
III) The set of rational numbers
The whole numbers comprise 0 and the positive integers. 
is a negative integer, so it belongs to the set of integers, but not the set of whole numbers. Also, all integers are rational by definition, since every integer can be expressed as the quotient of two integers (the integer divided by 1, for example). Therefore, 
is an integer and a rational number, but not a whole number, and the correct response is two.
The whole numbers comprise 0 and the positive integers.
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; 
, 
, and 
are distinct integers.
Which of the following could be equal to 
?
Which of the following could be equal to
We need to find ways to factor 32 such that the three factors are different, and then find the sum of those factors in each case.
32 can be factored as the product of three integers in five differrent ways:
I) 
II) 
III) 
IV) 
V) 
Of the five ways, only the second and third involve three distinct factors.
In Case II, the sum of the factors is
.
In Case III, the sum of the factors is
.
The only possible correct response is 19.
We need to find ways to factor 32 such that the three factors are different, and then find the sum of those factors in each case.
32 can be factored as the product of three integers in five differrent ways:
I)
II)
III)
IV)
V)
Of the five ways, only the second and third involve three distinct factors.
In Case II, the sum of the factors is
In Case III, the sum of the factors is
The only possible correct response is 19.
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.
, 
, and 
are integers; they may or may not be distinct.
Which of the following cannot be equal to 
?
Which of the following cannot be equal to
We look for ways to write 20 as the product of three integers, then we find the sum of the integers in each situation. They are:
Sum: 
Sum: 
Sum: 
Sum: 
9, 10, and 13 are among the possible sums, but 15 is not, so that is the correct response.
We look for ways to write 20 as the product of three integers, then we find the sum of the integers in each situation. They are:
Sum:
Sum:
Sum:
Sum:
9, 10, and 13 are among the possible sums, but 15 is not, so that is the correct response.
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.
and 
are integers; they may or may not be distinct.
Which of the following could be equal to 
?
Which of the following could be equal to
20 can be factored as:
I) 
II) 
III) 
The positive difference of the factors can be any of:
Of the four choices, only 8 is possible.
20 can be factored as:
I)
II)
III)
The positive difference of the factors can be any of:
Of the four choices, only 8 is possible.
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To how many of the following sets does the number belong?
I) The set of whole numbers
II) The set of integers
III) The set of rational numbers
To how many of the following sets does the number
I) The set of whole numbers
II) The set of integers
III) The set of rational numbers
The whole numbers comprise 0 and the positive integers. is a negative integer, so it belongs to the set of integers, but not the set of whole numbers. Also, all integers are rational by definition, since every integer can be expressed as the quotient of two integers (the integer divided by 1, for example). Therefore, is an integer and a rational number, but not a whole number, and the correct response is two.
The whole numbers comprise 0 and the positive integers.
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.
and are integers; they may or may not be distinct.
Which of the following could be equal to ?
Which of the following could be equal to
20 can be factored as:
I)
II)
III)
The positive difference of the factors can be any of:
Of the four choices, only 8 is possible.
20 can be factored as:
I)
II)
III)
The positive difference of the factors can be any of:
Of the four choices, only 8 is possible.
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How many subsets with three elements does the set 
have?
How many subsets with three elements does the set
The number of ways to select three elements from a set of eight is the number of combinations of three elements chosen from eight:
The number of ways to select three elements from a set of eight is the number of combinations of three elements chosen from eight:
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Which of the following numbers is a rational number but not an integer?
Which of the following numbers is a rational number but not an integer?
is a well-known irrational number and is not the correct choice.
. The square root of an integer is rational only if it is itself an integer; calculation yields a result of 1.414... This is not the correct choice.
, since 
. This is an integer and is not the correct choice.
, so, by definition, 
. This is rational and is therefore the correct choice.
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Which of the following sets does 
belong to?
(a) Whole numbers
(b) Integers
Which of the following sets does
(a) Whole numbers
(b) Integers
The set of whole numbers comprises 0 and the so-called natural, or counting, numbers; that is, it is the set 
. 
is not one of these numbers.
The set of integers comprises these numbers as well as their (negative) opposites; that is, it is the set 
. 
is one of these numbers.
The set of whole numbers comprises 0 and the so-called natural, or counting, numbers; that is, it is the set
The set of integers comprises these numbers as well as their (negative) opposites; that is, it is the set
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Which number is prime?
Which number is prime?
A prime number is a positive integer which has exactly two factors - 1 and the number itself. 36, 38, and 39 can each be shown to have at least one other factor:
, so 2 and 18 are factors of 36, making 36 not prime.
, so 2 and 19 are factors of 38, making 38 not prime.
, so 3 and 13 are factors of 39, making 39 not prime.
37, however, cannot be evenly divided by any number except for 1 and itself, as can be seen below:
We do not need to go higher, since any higher possible factors have square greater than 37. 37 has been proved a prime number.
A prime number is a positive integer which has exactly two factors - 1 and the number itself. 36, 38, and 39 can each be shown to have at least one other factor:
37, however, cannot be evenly divided by any number except for 1 and itself, as can be seen below:
We do not need to go higher, since any higher possible factors have square greater than 37. 37 has been proved a prime number.
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How many integers are in this set?
How many integers are in this set?
An integer is an element of the set of numbers
that is, the so-called natural, or "counting", numbers, their (negative) opposites, and 0. 
, 2, 
, and 1 are elements of this set; 
and 
are not. The correct response is 4.
An integer is an element of the set of numbers
that is, the so-called natural, or "counting", numbers, their (negative) opposites, and 0.
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The set of real numbers is divided into several subsets including positive numbers and negative numbers, prime numbers and composite number, rational numbers and irrational numbers, etc. For the following questions, select the answer that is a member of the stated subset.
Which number is prime?
The set of real numbers is divided into several subsets including positive numbers and negative numbers, prime numbers and composite number, rational numbers and irrational numbers, etc. For the following questions, select the answer that is a member of the stated subset.
Which number is prime?
A prime number is a number that is evenly divisible by 
, the number itself, and no other number.
is the only number that is not evenly divisible by a number other than 
and the number itself.
A prime number is a number that is evenly divisible by
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The set of real numbers is divided into several subsets including positive numbers and negative numbers, prime numbers and composite number, rational numbers and irrational numbers, etc. For the following questions, select the answer that is a member of the stated subset.
Which number is prime?
The set of real numbers is divided into several subsets including positive numbers and negative numbers, prime numbers and composite number, rational numbers and irrational numbers, etc. For the following questions, select the answer that is a member of the stated subset.
Which number is prime?
A prime number is a number with with no integer divisors, or factors, other than 
and the number itself. The prime factors of 
are 
and 
. (For technical reasons, 
is not considered prime and the number itself is not considered a factor.) The prime factors of 
are 
and 
. The prime factor of 
is 
. 
, however, cannot be divided evenly by any integer other than 
and 
thus is prime.
A prime number is a number with with no integer divisors, or factors, other than
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