Factoring of a number is the number decomposition into prime factors.

If we start from a composite number, by definition we can get your “decomposition” in factors less than the same number, these factors may, in turn, composite numbers or prime. If they are composites, We decompose in the same way in order to decomposition in a prime factors .

Example:

factors of 12
El 12 can be decomposed in 12 = 3 · 4, and in turn 4 = 2 · 2 therefore 12 = 3 · 2 · 2
El 12 can be decomposed too in 12 = 2 · 6, and in turn 6 = 2 · 3 therefore 12 = 2 · 2 · 3

factors of 40
2 ways to obtain factors of 40

factors of 600
ways to obtain factors of 600

whatever the factorization, in the end the prime factors are the same because there are unique possible decomposition in prime numbers.

Knowing this property, the more comfortable procedure to factoring is to use always the primes to split and the quotient to decompose.

¿HOW TO FACTORING?

Check if the number is divisible by 2; if it is, we make the Division and test with the quotient again with 2; if it’s not divisible we will see if it’s divisible by 3; and will repeat the same procedure than before. This will be repeated until the quotient it’s 1.

factors of 600 2
Factors of 600

Factors of: 28, 70, 32 y 210:

factors of 28,70,32
factors of 210

FACTORING FOR CALCULATING G.C.D. and L.C.M.

Factoring is also used to calculate the greatest common divisor (GCD) and least common multiple (LCM)

gcd of two numbers is the result of the common factors powered to smallest exponent.

lcm of two numbers is the result of the common and not common factors powered to highest exponent.

Example:
Calculate the gcd of 72 and 50

72 = 23 · 32

50=· 52

gcd(72,50) = 2

Calculate the lcm of 72 and 50

lcm(72,50) = 23 · 32 · 52 = 1800

We can use factoring to find out if a number is prime, as this will be prime when factoring only has as its prime factors 1 and the same.

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