What's an easy shorcut for factoring out the gcf?

Question

I find factoring out the gcf very hard so can someone tell me a shortcut or an easier way to do it???

Answer 1

The Greatest Common Factor (GCF) is the largest (greatest) factor (a factor multiplied by another factor makes a term) that is common (that is, in both) to both values.

 

When expressions are expressed with variables, it is rather easy.

  The GCF of:

           abcd

           acd

       is

           acd

       so, the expression

           abcd + acd

       may be written as:

           acd(b+1)             [note: Distributive Property]

 

This continues to be rather easy when expressions contain powers of variables:

    The GCF of:

           a³b²cd

           abde

       is

           abd

       so, the expression

           a³b²cd + abde

       may be written as:

           abd(a²bc+e)

 

Actually, the same process works with numbers, but we must find the prime factors of each number (The Sieve of Eratosthenes is an algorithm) then include 1.

 

The GCF of:

       72               1*2*2*2*3*3

       44               1*2*2*11

    is

       4                 1*2*2

 

The GCF of:

      11                 1*11

      13                 1*13

   is

       1                    1

 

To find the prime factors of a number, we may divide a number by each of the prime numbers until we have a remainder that is prime.  For example, to find the prime factors of 60, we start with 1 (not prime), then factor out a 2 if possible (2*30), again factor a 2 out of 30 if possible (2*2*15), then factor a 3 out of 15 if possible (2*2*3*5), then try to factor a 3 out of 5 again (sorry) and then factor a 5 (next prime) out of 5 (now done).  The result is 2*2*3*5=60.

 

To find the GCF between two numbers, list the factors obtained this way and identify the Greatest Common Factor.

Answer 2

Hey Ryle.

 

The best way to find the GCF for a set of numbers is to do a "prime factorization" for each number. The most common prime numbers you'll come across are 2, 3, 5, and 7. Factor out as many prime numbers as you can from each number then find which ones are redundant between the numbers you're finding the GCF for.

 

For example...

 

Say you want the GCF of 14, 42, and 56.

 

-First do the prime factorization of 14:

 

It's an even number, so 2 is an obvious prime factor:

 

14/ 2 = 7

 

7's also a prime number. Therefore, the prime factorization for 14 is:

 

2*7

 

-Next do the prime factorization of 42:

 

It's again an even number:

 

42/ 2 = 21

 

21 is divisible by 3.

 

21/ 3 = 7

 

7's again a prime number.

 

So the prime factorization of 42 is:

 

2*3*7

 

-Finally do this for 56.

 

56/ 2 = 28

 

28/ 2 = 14

 

14/ 2 = 7

 

7's the last prime number.

 

So the prime factorization for 56 is:

 

2*2*2*7

 

Now throw them all into some rows to compare:

 

14: 2*7

42: 2*3*7

56: 2*2*2*7

 

Now look at all of the prime factors and see which ones show up in all three of them.

 

Looks like just 2 and 7 show up in all of them.

 

2*7 = 14

 

This is your GCF.

 

Hope this helps.