Types of Numbers and Number Theory - GED Math

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 set with elements has subsets. Since , a set with 64 subsets has 6 elements.

, 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.

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.

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 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.

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.

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.

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.

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.

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 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 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.

Finding just one factor eliminates a number from the set of prime numbers. For a large number , begin with the smallest prime and exhaust all possible prime factors , , , , , etc.

with remaining

with remaining

with remaining

with remaining

with remaining

with remaining

For bonus points, explain why it is only necessary to test prime factors up to and including .

and so .

with remaining.

For any number , with remaining .

Thus, for a number to be a factor of , it must pair with another factor .

However, we have already exhausted all of the prime numbers so no prime factor exists.

A composite number is any number that isn't prime.

The remaining numbers are prime.

A positive number is any number that is greater than zero. is a negative decimal, is a negative quotient, is non-negative but it is not strictly to the right of zero on the number line.

, or pi, is a mathematical symbol for the ratio of a circle's circumference divided by twice its radius or

and, though irrational, is the only number listed that is strictly greater than zero. Irrational numbers are defined more rigorously later in the problem set.

This is a bit of a trick question. is an irrational number and the incorrect answers listed are all approximations of . However, every number listed can be expressed as a quotient of integers except . The square root of two is irrational as is the square root of any prime number.

A rational number is any number that can be expressed in the form where and are integers.

One of the answers is already in that form: is the ratio of two integers. And the repeating decimal is the decimal equivalent of that same ratio of 1:3.

is equal to 27 itself a rational number.

But cannot be written as the ratio of two integers, so it is an irrational number.

A rational number is a number that can be expressed in the form where both and are integers.

which is not only rational but an integer.

. Any square root of a prime number, the number 2 here, is irrational.

Finally, any square root of a negative number is complex and not real.