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# What terminology is related to factors and factoring?

There are several mathematical terms that include the word "factor" or are otherwise related to factoring.

A factor of a given number is an integer that evenly divides (without remainder) into that given number. Factors are usually assumed to be only positive integers unless the given number is negative. Factors are also known as divisors.

Example:

The factors of 6 are 1, 2, 3, and 6.

The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.

The factors of -15 are ±1, ±3, ±5, and ±15. A factor pairis a pair of factors that when multiplied together equal the given number. Example:

The factor pairs of 21 are 1 x 21 and 3 x 7.

The factor pair of 23 is 1 x 23.

The factor pairs of -14 are 1 x -14, 2 x -7, -1 x 14, and -2 x 7.

The proper factors of a given number are all the factors of the given number except the given number itself and sometimes 1. If proper factors are distinguished from proper divisors, proper factors exclude 1, but proper divisors include 1. Often, the term proper factors will be used for both, so check which is meant.

Example:

The proper factors of 14 are 2 and 7; the proper divisors of 14 are 1, 2, and 7.

There are no proper factors of 19; the proper divisor of 19 is 1.

A prime factor is a factor that is also a Prime number. Prime factors are not necessarily distinct; there can be multiple occurrences of the same prime factor. Sometimes, the term prime factors is used to mean distinct prime factors. The distinct prime factors of a given number is the list of each distinct (or different) prime factor for that given number. Although -1 is not a prime number, it is included in the list of prime factors when necessary to represent that a number is negative.

Example:

The prime factors of 24 are 2, 2, 2, and 3; the distinct prime factors of 24 are 2 and 3.

The prime factors of 30 are 2, 3, and 5; the distinct prime factors of 30 are 2, 3, and 5.

The prime factors of -50 are -1, 2, 5, and 5; the distinct prime factors of -50 are -1, 2, and 5. The least prime factor is the smallest factor greater than 1, which will be a prime number. The greatest prime factor is the largest factor that is prime. Example:

The least prime factor of 24 is 2; the greatest prime factor of 24 is 3.

The least prime factor of 30 is 2; the greatest prime factor of 30 is 5.

The least prime factor of 73 is 73; the greatest prime factor of 73 is 73.

The prime factorization of a given number (also called the prime decomposition) is the unique representation of primes multiplied together that equal the given number. The index form of prime factorization is the prime factorization using exponents for repeated prime factors.

Example:

The prime factorization of 20 is 2 x 2 x 5; the index form of the prime factorization is 22 x 5.

The prime factorization of 27 is 3 x 3 x 3; the index form of the prime factorization is 33.

The prime factorization of -39 is -1 x 3 x 13.

Common factors are factors that two or more numbers have in common. A single number cannot have common factors; there must be at least two or more numbers. The greatest common factor(also known as a highest common factor) is the largest factor that two or more numbers have in common. The least common factor is the smallest factor that two or more numbers have in common, which will usually be 1.

Example:

The common factors of 4 and 6 are 1 and 2; the greatest common factor is 2.

The common factors of 12, 18, and 24 are 1, 2, 3, and 6; the greatest common factor is 6.

The common factors of 13 and 19 are only 1; the greatest common factor is 1.

Note: Some mathematics textbooks state that there is not a least common factor (referring to the existence of the terminology) because it is always 1. Some people erroneously use least common factor when they mean least common multiple. Some computer programming books define least common factor as the smallest common factor greater than 1 unless the only common factor is 1.

A factor tree is a diagram used to determine the prime factors of a composite number. It is constructed by finding a factor pair of the given number. Then, find the factor pairs of those numbers until all the factors are prime. Some numbers have several factor trees. It does not matter which factor pair is chosen to start the factor tree.

Example:

A factor tree of 48 is (using periods to adjust the spacing)

.........48

......../...\

......6 .x. 8

..../..\..../..\

...2 x 3 2 x 4

................/..\

...............2 x 2

Thus, the prime factors are 2, 2, 2, 2, and 3.

A factor tree of 60 is (using periods to adjust the spacing)

.........60

......../...\

......4 .x. 15

..../..\...../..\

..2 x 2 . 3 x 5

Thus, the prime factors are 2, 2, 3, and 5.

A factor rainbow is a diagram used to determine all the factors and factor pairs of a number. Start by making a factor pair of the number and 1. This pair of factors is joined by a curving line like a color in a rainbow. Then, check whether 2 goes into the number, then 3, then 4, and so on. Each factor pair is joined by a curving line within the previous factor pair. When finished, you have an ascending list of all the factors of the number and the set of factor pairs. Example:

A factor rainbow of 24 (using periods to adjust the spacing)

_______________

|........................|

1.......................24

Then, continue adding in the remaining factor pairs until finished.

__________________

| . _____________ . .|

| . | . ________ . | . .|

| . | . | . ___ . | . | . . |

| . | . | . | . | . | . | . . |

1..2..3...4...6..8..12..24

A factor string is a set of two or more factors, excluding 1, that when multiplied together equal the given number. Unlike prime factorization, these factors do not need to be prime factors. However, the longest factor string of a number is its prime factorization. The length of a factor string is the number of factors listed. Example:

A factor string of 18 is 3 x 6. it has a length of 2.

A factor string of 24 is 2 x 3 x 4; it has a length of 3.

A factor string of 120 is 2 x 3 x 4 x 5; it has a length of 4. A factor ladder, also known as a division ladder, is a method of determining the prime factors of a number. Start by dividing by the smallest prime number that will go into the number. Divide that result by the smallest prime number that will go into it. Continue until the result is a prime number. All of the prime numbers that were used as divisors and the final prime number result are the prime factors of the number. These instructions say to start with the smallest prime number because that it easier with very large numbers and is the more common way to do it, it does not actually matter in which order you divide by the prime numbers. Sometimes ladders are built downwards, while other times they are built upwards. Sometimes, a division ladder built upwards is known as a factor cake because each "layer" is "smaller" than the one beneath it - and the physical resemblance.

Example:

A division ladder of 72 built downwards:

2 ∕--72

2 ∕--36

2 ∕--18

3 ∕--9

3

The prime factors of 72 are 2, 2, 2, 3, and 3. The prime factorization of 72 is 23 x 32.

A division ladder of 90 built upwards:

5

3 ∕--15

3 ∕--45

2 ∕--90

The prime factors of 90 are 2, 3, 3, and 5. The prime factorization of 90 is 2 x 32 x 5. Study guides

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## A number a power of a variable or a product of the two is a monomial while a polynomial is the of monomials

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