What are prime numbers and factors; and how do you find them?
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A prime number is a number that cannot be divided by a whole number to get another whole number except for 1 and itself. Another way to think about it is that prime numbers are numbers that cannot be created by multiplying two whole numbers other than 1 and itself.
12 is definitely not prime because you van make it as 2×6 and 3×4. 7 is prime bevause there are no whole numbers that multiply it.
Because non-prime numbers can be written as a multiplication of two other whole numbers those numbers themselves can be prime or not prime. For example 12=3×4. 3 is prime but 4 isnt because 2×2 makes 4. 2 is prime but only because there are no numbers other than 1 smaller than it.
Therefore you can decompose any non prime into its prime factors. These are the prime numbers that when multiplied create that number. So the prime factors of 12 are 3×2×2. Etc.
Factors are numbers of which a other number is divisible by. For exams 8 is divisible by 4 and 2 so 8 has factors 4 and 2. As well as 1 and 8. So the factors of 8 are 1 2 4 and 8.
Prime numbers are numbers that have only
Two factors. 1 and itself. For example 13. It’s only wholly divisible by 1 and 13.
Factors (numbers only): A natural number that multiplies by another natural number to equal a natural number.
Prime (numbers only): An natural number that only has 1 & itself as factors.
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2 & 5 are factors of 10 (2•5=10). 10 is not prime (composite) as it has 1,2,5,10 as factors.
1 & 7 are the only factors of 7 (1•7=7). 7 is prime as it only has 1 & 7 as factors.
A prime number is a number greater than 1 which is only divisible by 1 and itself. That is to say if you divide a prime by any whole number that isn’t 1 or itself, the result **won’t** be a whole number. The first few primes are 2, 3,5,7,11,13,17… if you try and divide any of these numbers by anything other than 1 or the number itself the result not be a whole number.
There isn’t really a shortcut or easy to find primes other than just dividing by all possible numbers and checking if the result is whole or not. There’s a few minor shortcuts you can use to quickly eliminate some factors, but not all. (I’ll list them under factors)
Factors are the whole numbers by which a whole number is divisible. For example you can divide the number 12 by 1,2,3,4,6 and 12 and get a whole number as a result, so 1,2,3,4,6 and 12 are the factors of the number 12.
To find factors, again there is no shortcut to find all the factors of a number other than just trying out numbers, but as mentioned there are some shortcuts:
Every number that has 3 as a factor will have a sum of digits divisible by 3. For example 96 has a sum of digits of 9+6 = 15 and 15 is divisible by 3, so 96 is also. (This can obviously be repeated for large numbers, e.g. if you wanted to know if 654987318756491587695 is divisible by 3 or not you sum the digits 6+5+4+9+8+7+3+1+8+7+5+6+4+9+1+5+8+7+6+9+5 = 123. Then just repeat 1+2+3 = 6 and 6 is divisible by 3, so we now calculated entirely in our head that 654987318756491587695 is a multiple of 3)
Every number that has 2 as a factor must have an even number as last digit (2,4,6,8,0)
Every number that has 5 as a factor must have either 5 or 0 as a last digit
Every number that has 10 as a factor must have 0 as a last digit
Every number that has 100 as a factor must have 00 as the two last digits (this pattern just continues for 1000, 10 000, 100 000 …..)
Other than those shortcuts there is no real way to determine a numbers factors other than trying it out.
One last thing of note is that the factors of a factor are always also factors of the original number, e.g. 100 has the factor 25 (25*4 = 100) and 25 has the factor 5 (5*5 = 25). This means 5 must also be a factor of 100.
You’ll notice you can combine these two to say that a prime number is a number which only has 1 and itself as a **factor.**
There is also another interesting bit about factors, which is called “prime factors”. Every positive whole number that exists can be expressed as a product of prime numbers. More importantly a **unique** product of primes. In contrast to regular factors, when talking about prime factors you can list the same factor twice. An example:
12 has the prime factorization 2*2*3
This means 2^(2) * 3 is the **only** possible way to multiply prime numbers together and get 12.
Take a number of pieces (pebbles, chess pawns, domino pieces, whatever, or just imagine it in your head) and try to lay them out in a rectangular grid. You will notice that for some numbers, like 5 or 13, the only possible “rectangle” is to lay them out in one long line. Those numbers are called “prime numbers”, the numbers that can form a proper rectangle are “composite numbers”.
Now take any composite number and form a rectangle with it. The two sides of the rectangle will each be themselves either a prime or a composite number. For both numbers, if it’s a prime number, note it down, if it’s a composite number form a rectangle again and repeat. You will end up with a bunch of prime numbers; that’s the prime factorization of your original number.
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