I. DefinitionsFactors
"Factors" are the numbers you multiply together to get another number. Another way of saying it is that the factor of a number is a number that divides the given number completely without any remainder.
2 x 3 = 6 (here 2 and 3 are factors of 6)
Factor Pairs are 2 whole numbers that can be multiplied to get a certain product. For example, the factor pairs of 12 are (1,12), (2,6) and (3,4). From these factor pairs we can see that the factors of 12 are 1, 2, 3, 4, 6, 12.
Multiples
Multiples are the numbers you get when you multiply a certain number by another number. Another way of saying it is that a multiple of a number is the product of the number and a natural number. For example, the multiples of 12 are 12, 24, 36, 48 and so on.
Prime Numbers
A prime number is a whole number greater than 1 which can only be divided evenly (not producing a remainder) by 1 and itself. Stated another way, A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. It is a number that has only two factors, itself and one. A natural number greater than 1 that is not prime is called a composite number.
For example, 5 is prime because the only ways of writing it as a product are 1 × 5 or 5 × 1. However, 6 is composite because it is the product of two numbers (2 × 3) that are both smaller than 6.
Prime numbers are {2, 3, 5, 7, 11, 13, 17...}.
Composite numbers are {4, 6, 8, 9, 10, 12, 14...}
Note that the only even prime number is 2.
Primes are central in
number theory because of the
fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique
up to their order.
II. Finding Factors and MultiplesA. Finding All The Factors Of A NumberMultiplication Method
Example: Find the factors of 24
Step 1:
In order to find the factors of 24 using multiplication, we need to check what pairs of numbers multiply to give us 24, so we need to divide 24 by natural numbers starting from 1
Step 2:
We write that particular number along with its pair and make a list as shown in the figure given below. As we check and list all the numbers, we automatically get the other pair factor along with it. For example, starting from 1, we write 1 × 24 = 24 and 2 × 12 = 24 and so on. Here, (1, 24) forms the first pair, (2, 12) forms the second pair and the list goes on as shown.
To find the multiples of a number simply multiply the given number by the natural numbers starting with 1.
Example:
What are the first four multiples of 7?
7 x 1 = 7
7 x 2 = 14
7 x 3 = 21
7 x 4 = 28
So 7, 14, 21, 28 are the first four multiples of 7
C. Identifying Prime Numbers
You can use the factorization (finding the factors of a number) to see if a number is prime. This works fine for small numbers but is difficult for large numbers. There are some methods you can use
here for large numbers
D. Prime Factorization
Prime factorization is the decomposition of a composite number into a product of its prime numbers.
Many algorithms have been devised for determining the prime factors of a given number. The simplest method of finding factors is trial division using a factor tree or division method:
For example, let's do the prime factorization of 24.
Step 1
Find any two numbers that multiply together to make the number you started with. You can use any two you can think of, but a prime number will make your work easier. One good strategy is to try dividing the number by 2, then 3, then 5, working your way up through the prime numbers until you find one that divides evenly.
Let's divide by 2 to get 24 = 2 x 12
24
/\
2 12
Step 2
Since 12 is a composite number, it will need to be factored.
24
/\
2 12
/\
2 6
Step 3
Since 6 is a composite number it will need to be factored.
/\
2 3
Result
Now that every factor is prime we have our answer
24 = 2 x 2 x 2 x 3
We can use exponents (see real numbers)
24 = 23 x 3
III. Miscellaneous
A. Divisibility Rules
