How do you factor?

Basics of how to factor

• Any non-prime number can be represented as a product: 6 is 2 x 3, 64 is 8 x 8, and so on. This is basically how you factor by representing a number as a product.
• Prime numbers are by definition Countingthat can only be divided by 1 and the number itself. Now, before you test whether 23 is a prime number by dividing it by 1 and by 23, each number is divisible by 1 and itself. Leave out the exact definition of the prime number and use the slang idea as an alternative, a number that you cannot divide by anything without a remainder.
• You either need to know the prime numbers at least from 1 to 100 by heart, or one Prime numbers table on hand to decide whether a number is prime, because when factoring is done, you don't have time to test all the numbers.

Safe way to factorize

Using the number 2520 as an example, you can see how to factorize.

1. Divide the number 2520 by the smallest known prime number (not 1 of course). You get 1260. So 2520 = 2 x 1260.


Prime numbers 1-100 - this is how you determine them with a system

If you are to calculate the prime numbers from 1-100, you can do this after the sieve of ...

2. Divide 1260 by 2 again, you get 630, so 2520 = 2 x 2 x 630.
3. Divide 630 by 2 and you see that 2560 = 2 x 2 x 2 x 315.
4. Since 315 cannot be divided by 2 without a remainder, divide 3 by the next prime number. 315: 3 = 105, so 2560 = 2 x 2 x 2 x 3 x 105.
5. Now 105 is again divided by 3 and you get 2560 = 2 x 2 x 2 x 3 x 3 x 35.
6. Since 35 is not divisible by 3, you now have to divide by 5 to get 2520 = 2 x 2 x 2 x 3 x 3 x 5 x 7. So you've factored in prime factors because all the numbers in the product are prime numbers. You can also write this with exponents. So 2520 = 2³ x 3² x 5 x 7.

Divisibility rules help when factoring

As you have seen in the example, you have to divide. It is helpful if you know a few rules of divisibility. This makes it easier to decide whether a number can be divided by certain prime factors:

• A number is divisible by 2 if the last digit is divisible by 2, i.e. 2, 4, 6, 8, 0.
• A number is divisible by 3 if the checksum is divisible by 3.
• A number is divisible by 5 if the last digit is a 5 or a 0.
• Even if 4 and 10 are not prime numbers. Knowing that a number is divisible by 4 if the last two digits are divisible by 4 and that a number is divisible by 10 if there is a 0 at the end will always help you.

Factoring Tricks

You do not have to decompose directly into prime numbers; you also factor if you first decompose into arbitrary factors and then decompose them further. Again with example 2560:

1. 2560 has a zero at the end, so 2520 = 10 x 252.
2. 256 is an even number, i.e. divisible by 2 252 = 2 x 126, so 2520 = 10 x 2 x 126.
3. Since 126 is divisible by 2 and 10 is 2 x 5, the following applies: 2520 = 2 x 5 x 2 x 2 x 63.
4. 63 is divisible by 3, which is 3 x 21 and 21 is 3 x 7. So 2520 = 10 x 252 = 2 x 5 x 2 x 16 = 2 x 5 x 2 x 2 x 63 = 2 x 5 x 2 x 2 x 3 x 3 x 7.
5. Sort the numbers according to size and you have 2520 again = 2 x 2 x 2 x 3 x 3 x 5 x 7 = 2³ x 3² x 5 x 7.

The first method is very safe and can be calculated quite stubbornly according to a scheme, but it often takes a long time. The second method requires some feeling for numbers and a good concentration so that you don't forget any factors. Both methods are used to factor correctly.