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عدد المساهمات : 57
تاريخ التسجيل : 26/12/2009

مُساهمةموضوع: العامل المشترك الاعلى   الخميس مارس 25, 2010 8:47 pm

LCM (Least Common Multiple) and
GCF (Greatest Common Factor)
To find either the Least Common Multiple (LCM) or Greatest Common Factor (GCF) of two numbers, you always start out the same way: you find the prime factorizations of the two numbers. Then you put the factors into a nice neat grid of rows and columns, and compare and contrast and take what you need.
• Find the GCF and LCM of 2940 and 3150.
First, I need to factor each value:



My prime factorizations are: Copyright ©️ Elizabeth Stapel 1999-2009 All Rights Reserved
2940 = 2 × 2 × 3 × 5 × 7 × 7
3150 = 2 × 3 × 3 × 5 × 5 × 7
I will write these factors out, all nice and neat, with the factors lined up according to occurrance:

This orderly listing, with each factor having its own column, will do most of the work for me
The Greatest Common Factor, the GCF, is the biggest number that will divide into (is a factor of) both 2940 and 3150. In other words, it's the number that contains all the factors common to both numbers. In this case, the GCF is the product of all the factors that 2940 and 3150 have in common.
Looking at the nice neat listing, I can see that the numbers both have a factor of 2; 2940 has a second copy of the factor 2, but 3150 does not, so I can only count the one copy toward my GCF. The numbers also share one copy of 3, one copy of 5, and one copy of 7.

Then the GCF is 2 × 3 × 5 × 7 = 210.
On the other hand, the Least Common Multiple, the LCM, is the smallest number that both 2940 and 3150 will divide into. That is, it is the smallest number that contains both 2940 and 3150 as factors, the smallest number that is a multiple of both these values. Then it will be the smallest number that contains one of every factor in these two numbers.
Looking back at the listing, I see that 3150 has one copy of the factor of 2; 2940 has two copies. Since the LCM must contain all factors of each number, the LCM must contain both copies of 2. However, to avoid over duplication, the LCM does not need three copies, because neither 2940 nor 3150 contains three copies.
This fact often causes confusion, so let's spend a little extra time on this. Consider two smaller numbers, 4 and 8, and their LCM. The number 4 factors as 2 × 2; 8 factors as 2 × 2 × 2. The LCM needs only have three copies of 2, in order to be divisible by both 4 and 8. That is, the LCM is 8. You do not need to take the three copies of 2 from the 8, and then throw in two extra copies from the 4. This would give you 32. While 32 is a common multiple, because 4 and 8 both divide evenly into 32, 32 is not the LEAST (smallest) common multiple, because you over-duplicated the 2s when you threw in the extra copies from the 4. Again, let the nice neat listing keep track of things when the numbers get big.
So, my LCM of 2940 and 3150 must contain both copies of the factor 2. By the same reasoning, the LCM must contain both copies of 3, both copies of 5, and both copies of 7:

Then the LCM is 2 × 2 × 3 × 3 × 5 × 5 × 7 × 7 = 44,100.
By using this "factor" method of listing the prime factors neatly in a table, you can always easily find the LCM and GCF. Completely factor the numbers you are given, list the factors neatly with only one factor for each column (you can have 2s columns, 3s columns, etc, but a 3 would never go in a 2s column), and then carry the needed factors down to the bottom row.
For the GCF, you carry down only those factors that all the listings share; for the LCM, you carry down all the factors, regardless of how many or few values contained that factor in their listings.
________________________________________
• Find the LCM and GCF of 27, 90, and 84.
First, I need to find the prime factorizations:

Then I will list these factorizations neatly:

Then the GCF (being the product of the shared factors) and the LCM (being the product of all factors) are given by:

Then the GCF is 3 and the LCM is 3,780.
• Find the GCF and LCM of 3, 6, and 8.
First I factor the numbers and list their prime factorizations:

Then my GCF and LCM are given by:

Note that 3, 6, and 8 share no common factors. While 3 and 6 share a factor, and 6 and 8 share a factor, there is no prime factor that all three of them share. Since 1 divides into everything, then the greatest common factor in this case is just 1. When 1 is the GCF, the numbers are said to be "relatively" prime; that is, they are prime, relative to each other.
Then the GCF is 1 and the LCM is 2 × 2 × 2 × 3 = 24.
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The GCF doesn't come up that much at this stage in mathematics, though some books use it for factoring polynomial expressions by having the student find the GCF of all the terms in the polynomial and divide this value out of every term. But the LCM comes up every time you need to find a common denominator for fractions. The factor technique I demonstrated above works even for polynomial fractions. (The other method for finding the LCM, the "listing" method, will not work for polynomials, which is why you need to learn the factor method.) If you need to find the LCM of two (or more) polynomials, you can do the exact same procedure as above:
• Find the LCM of x3 + 5x2 + 6x and 2x3 + 4x2.
First I factor the polynomials: x3 + 5x2 + 6x = x(x2 + 5x + 6) = x(x + 2)(x + 3), and 2x3 + 4x2 = 2x2(x + 2). Then I list these factors out, nice and neat:

Then the LCM is the product of one entry from each column:

I take two copies of "x", because 2x3 + 4x2 contains two copies. I don't need three copies of "x", because neither polynomial contains three copies. I need only one copy of x + 2, because neither polynomial contains more than just the one copy. I need to account for the 2 from the second polynomial and the x + 3 from the first polynomial.
The LCM is 2x2(x + 2)(x + 3).
By the way, while you always multiply the factors together when you're finding the common denominators for regular fractions, you almost always want to leave the common denominators for polynomial fractions in factored form. That is, you'll need to multiply and simplify across the top, but don't multiply anything together across the bottom (in this case, don't multiply the x + 2 and the x + 3 to get x2 + 5x + 6). Remember: you'll still need to try to reduce the polynomial fraction when you're done simplifying across the top, so you'll need the bottom in factored form in the end, anyway.
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