User:MathMan64~enwikibooks/Arithmetic/Factoring

= Factoring =

Factors, multiples, divisors, and divisibility
The four terms above apply only to whole numbers.


 * 2 is a factor of 6, because 2 times 3 is 6.
 * 6 is a multiple of 2 for the same reason.
 * 2 is a divisor of 6 because 6 divided by 2 is 3 with no remainder.
 * 6 is divisible by 2 for the reasons above.

These four statements all say the same thing from different perspectives. The terminology used below refers to the smaller numbers as factors, the larger number as being divisible by each of the factors.

Divisibility
There are several ways to tell if one number is divisible by another.

Multiplication
If the numbers are small, a multiplication fact should come to mind.
 * 56 is divisible by 7 because 7 times 8 is 56.
 * 27 is not divisible by 4 because 4 times 6 is 24 and 4 times 7 is 28. Since there is no whole number between 6 and 7, there is no multiple of 4 between 24 and 28.

Division
A way to show divisibility that always works is to divide and see whether or not there is a remainder.


 * 42 is divisible by 14 because 42 divided by 14 is 3 with no remainder.
 * 72 is not divisible by 13 because 72 divided by 13 is 5 with a remainder of 7.
 * 5391 is not divisible by 61 because on a calculator, 5361 divided by 61 is 88.38 (rounded.)
 * 5141 is divisible by 53 because on a calculator 5141 divided by 53 is exactly 97.

Divisibility tests
The most useful way to tell if one number is divisible by another for the following three particular divisors are called the divisibility tests.

Two
Divisibility by 2 can be seen by looking at the one's digit. If the number is even, with a one's digit of 0, 2, 4, 6, or 8; then it is divisible by 2.
 * It is seen that that 78 is divisible by 2, by looking at the one's digit. It is 8, thus 78 is an even number and thus divisible by 2.
 * It is seen that 9105 is not divisible by 2, by looking at the one's digit. It is 5, thus 9105 is an odd number, and not divisible by 2.
 * It is seen that 392,845,160 is divisible by 2, by looking at the one's digit. Since it is 0, the number is divisible by 2.

Five
Divisibility by 5 can be seen by looking at the one's digit. If the number has a one's digit of 0 or 5, then it is divisible by 5. (Think of money in nickels.)
 * It is seen that 78 is not divisible by 5, by looking at the one's digit. It is 8.  Since it is not 0 or 5, 78 is not divisible by 5.
 * It is seen that 9105 is divisible by 5, by looking at the one's digit. It is 5.  Thus 9105 is divisible by 5.
 * It is seen that 392,845,160 is divisible by 5, by looking at the one's digit. Since it is 0, the number is divisible by 5.

Three
Divisibility by 3 is a little more difficult. It takes an extra step. Instead of looking just at the one's digit, all the digits must be added up. If the sum of the digits is divisible by 3, then the original number is divisible by 3. When the sum of the digits is a one-digit number, if the sum is 3, 6, or 9; the original number is divisible by 3. If the sum of the digits is any other one-digit number, then the original number is not divisible by three.
 * 81 can be shown to be divisible by 3, by adding up the digits of 81. The sum of the digits 8 + 1 is 9, so 81 is divisible by 3.
 * 71 can be shown to be not divisible by 3, by adding up the digits of 71. The sum of the digits 7 + 1 is 8, so 71 is not divisible by 3.

If the sum of the digits is greater than 9, the digits of the sum can be added again to see whether or not 3, 6, or 9 results.
 * It can be shown that 78 is divisible by 3, by adding up the digits. The sum of 7 + 8 is 15, and 1 + 5 is 6; thus 78 is divisible by 3.
 * It can be shown that 9105 is divisible by 3, by adding up the digits. The sum of 9 + 1 + 0 + 5 is 18, and 1 + 8  is 9; thus 9105 is divisible by 3.
 * It can be shown that 392,845,160 is not divisible by 3, by adding up the digits. The sum of 3 + 9 + 2 + 4 + 5 + 1 + 6 + 0 is 38, and 3 + 8 is 11, and 1 + 1 is 2; therefore 392,845,160 is not divisible by 3.

Prime and composite
Most whole numbers have several factors.


 * As stated above 2 and 3 are factors of 6. But 1 and 6 are also factors of 6.  The only factors of 6 are 1, 2, 3, and 6 because no other number divides into 6 with no remainder.
 * The factors of 12 are 1, 2, 3, 4, 6, and 12. Notice that these factors occur in pairs whose product is the number being factored: (1 and 12, 2 and 6, 3 and 4.)
 * 1 and 19 are the only factors of 19. No other numbers divide into 19 evenly.
 * The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100. All these factors occur in pairs except for 10, since 100 divided by 10 is 10 itself.

Notice in these examples, that every whole number is a factor of itself, and that 1 is a factor of every number. Some numbers have only two factors. 1 and the number itself are the only factors of these numbers.

Definitions

 * Numbers with only two factors are prime numbers. The only factors of a prime are 1 and the number itself.
 * Numbers with more than two factors are composite numbers.
 * Numbers with less than two factors are neither prime nor composite. Zero and one are the only whole numbers that are neither prime nor composite.

Examples
The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

Composite
If a number is divisible by any number besides 1 and the number itself, then it is composite. If one other factor can be found, (any factor other than 1 and the number itself) then the number is shown to be composite. Trying primes as factors is all that is needed, but if it is obvious that a composite number is a factor, the original number is shown to be composite.
 * 124 is divisible by 2, so it is composite.
 * 125 is divisible by 5, so it is composite.
 * 126 is divisible by 3, so it is composite.
 * 140 is equal to 10 times 14, so it is composite.
 * 143 is not divisible by 2, 5, or 3. So we divide by 7, but it doesn't come out even.  So we divide by 11, and it does divide with no remainder.  So 143 is composite.

Prime
If a number is not divisible by any number other than one and the number itself, then it is prime. This is much more difficult. It must be shown that the number is not divisible by any other factor. Only smaller numbers need to be checked. Trying to divide by smaller primes will be sufficient. In fact, it's only necessary to use the primes which when multiplied by themselves give a product that is smaller that the original number.
 * 43 is not divisible by 2, 5, or 3. And 7 times 7 is larger than 43.  So 43 is prime.
 * 173 is not divisible by 2, 5, or 3. When 173 is divided by 7, 11, and 13; none of the divisions result with no remainder.  And 17 times 17 is 289, which is larger than 173.  So 173 is prime.

Two rules

 * The rule of 48 is that any number less than 48 only needs to be checked for divisibility by 2, 3 and 5. This works because 7 is the next prime, and 7 times 7 is 49.
 * The rule of 120 is that any number less than 120 only needs to be checked for divisibility by 2, 3 and 5, and then try dividing by 7. This works because 11 is the next prime, and 11 times 11 is 121.

Prime factorization
Factoring a number into primes is an important skill in working with the fractions of arithmetic. Reducing and finding a common denominator when the numbers in the fraction are large, is made easier as the numbers are factored. The concepts used in working with numeric fractions in this way, are concepts important when working with fractions in algebra.

The fundamental theorem of arithmetic states that every whole number can be factored into the product of primes in only one way. The order in which the factors are listed does not make any difference. The easiest way to be consistent in listing the factors is to have them in order with the small ones first.

The process of drawing a tree diagram will be useful as large numbers are considered. In mathematics, trees are often drawn upside down, with the trunk (or root) at the top and the branches going down from there.

There are three ways to approach the factoring process.

Multiplication
When the number is less than 100, consider whether or not it is the product of a common multiplication fact.


 * Find the prime factorization of 35.


 * The prime factorization of 35 is 5 &times; 7.

In the case of a prime number, the factorization is written with only the one number and no multiplication sign.
 * When trying to factor 43, there is no example in the multiplication table that equals 43; none of the factors 2, 5 or 3, are divisors of 43; and 43 is less than 48; so 43 is prime.


 * The prime factorization of 43 is 43.

With more than two factors, a larger tree and more steps are needed. The factoring process continues until the numbers at the bottom of the tree are all prime.


 * Find the prime factorization of 72.


 * The prime factorization of 72 is 2 &times; 2 &times; 2 &times; 3 &times; 3 or 23 &times; 32.


 * Find the prime factorization of 16.


 * In either case, the prime factorization of 16 is 2 &times; 2 &times; 2 &times; 2 or 24.

Divisibility tests
The divisibility tests are very useful when there is not an easily remembered multiplication fact with the product that needs factoring.


 * Find the prime factorization of 111.


 * 111 is not divisible by 2 or 5, but it is divisible by 3. Division of 111 by 3 gives 37.
 * The prime factorization of 111 is 3 &times; 37.


 * Find the prime factorization of 72 using the divisibility tests.


 * The prime factorization of 72 is 2 &times; 2 &times; 2 &times; 3 &times; 3 or 23 &times; 32.


 * Find the prime factorization of 2310.


 * The prime factorization of 72 is 2 &times; 3 &times; 5 &times; 7 &times; 11.

Division
When all else fails, division must be used.


 * Find the prime factorization of 2873.


 * The prime factorization of 2873 is 13 &times; 13 &times; 17 or 132 &times; 17.

Often combinations of the three factoring methods can be used.


 * Use all three methods to find the prime factorization of 11,730.


 * The prime factorization of 11,730 is 2 &times; 3 &times; 5 &times; 17 &times; 23.