# GCF factoring Question

• May 23rd 2008, 11:48 AM
cmf0106
GCF factoring Question
Regarding GCF factoring, can the same answer be derived from multiple number sets? Specifically fin the following problem:

$40a^5x+80a^5y-120a^5z$ & its answer 40a^5(x + 2y - 3z)

40 obviously is the GCF and goes into all three numbers. But what if one doesn't realize this initially and decides to approach it as such?

pulling out the terms
$40a^5x = 2^2*10$
$80a^5y=2^4*5$
$-120a^5z= 2^3*15$

then finding the lowest power of each variable and number. Will one still arrive at the same answer as if they used 40 as the GCF?
• May 23rd 2008, 12:14 PM
masters
This is true.

$40=2^3\cdot5$

$80=2^4\cdot5$

$120=2^3\cdot3\cdot5$

Use all the common factors with the least exponent.

Thus, we have $2^3\cdot5=40$

Do the same for the variables. Find the common variables and use the least exponent.
• May 23rd 2008, 12:20 PM
cmf0106
Yikes, you changed yours up a little bit in your example

Instead of what I used, 2^2 * 10 = 40 for the first term you used 2^3 * 5 was. So is answer is already wrong at this point, going with 2^2 * 10 = 40? If so how can I avoid future problems such as this?
• May 23rd 2008, 12:29 PM
masters
Quote:

Originally Posted by cmf0106
Yikes, you changed yours up a little bit in your example

Instead of what I used, 2^2 * 10 = 40 for the first term you used 2^3 * 5 was. So is answer is already wrong at this point, going with 2^2 * 10 = 40? If so how can I avoid future problems such as this?

You should factor your numbers into the product of primes.

$40=2^3\cdot5$, otherwise you may miss an exponent. You were lucky this time since you already had a $2^3$ as a factor of 120.
• May 23rd 2008, 12:36 PM
Reckoner
Quote:

Originally Posted by cmf0106
Yikes, you changed yours up a little bit in your example

Instead of what I used, 2^2 * 10 = 40 for the first term you used 2^3 * 5 was. So is answer is already wrong at this point, going with 2^2 * 10 = 40? If so how can I avoid future problems such as this?

When factoring numbers to find the GCF, you should factor everything completely into powers of prime numbers. 10 is not prime, because $10=2\cdot5$, but 2 and 5 are both prime. Once you have factored completely, then you can take the common factors, and their product will be your greatest common factor.
• May 23rd 2008, 12:39 PM
masters
Could not have said it better myself.