LCM and GCF

To find the Greatest Common Factor, I need to find all of the factors, prime and otherwise, of each of the two numbers. The best way I've found to do this is to find factor pairs; that is, I'll find one number that divides evenly into the number, and then do the division, which gives me the other factor in the pair. So, grabbing my calculator...

(I found these factor pairs by dividing the original numbers in my calculator by progressively larger values, starting at 2 and continuing until the answer after division was smaller than what I'd divided by.)

Now I list the factors in order, from least to greatest:

The largest value that is in both lists is 210, so this is the GCF.

Now I have to start listing the multiples of each of the original numbers, until I find a duplicate:

While I was making my list of multiples of 3150, I had to keep extending my list of multiples of 2940, until I finally found a duplicate. This duplicate, 44100, is the LCM.

First, I need to factor factor each of the numbers they've given me:

2)2940
 -----
2)1470
 -----
 3)735
  ----
 5)245
  ----
  7)49
   ---
     7

(This factorization process is pretty easy: divide by the smallest prime that goes in evenly, working your way through the primes until the answer to your division is itself a prime. Review this lesson if you need a refresher.)

Now I'll apply the same sequential-division process to 3150:

2)3150
 -----
3)1575
 -----
 3)525
  ----
 5)175
  ----
  5)35
   ---
     7

The factorizations can be read from the numbers along the outside of the sequential divisions, so my prime factorizations are:

I will write these factors out, all nice and neat, with the factors lined up according to occurrance:

2940: 2×2×3  ×5  ×7×7
3150: 2  ×3×3×5×5×7

This orderly listing, with each factor having its own column, will be doing most of the work for me.

The Greatest Common Factor, the GCF, is the biggest (that is, the "greatest") number that will divide into (that is, the largest number that 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 share.

So, to find the GCF, I just take all the factors that are in both factorizations:

2940: 2×2×3  ×5  ×7×7
3150: 2  ×3×3×5×5×7
----:----------------
 GCF: 2  ×3  ×5  ×7  = 210

Then the GCF is 2 × 3 × 5 × 7 = 210

On the other hand, the Least Common Multiple, the LCM, is the smallest (that is, the "least") 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* that is common to both these values. Therefore, it will be the smallest number that contains every factor in these two numbers.

So, to find the LCM, I just take all the factors from each of the columns in my factor grid:

So, to find the GCF, I just take all the factors that are in both factorizations:

2940: 2×2×3  ×5  ×7×7
3150: 2  ×3×3×5×5×7
----:----------------
 LCM: 2×2×3×3×5×5×7×7 = 44,100

Note that I took each column's factor; I did not use all the copies of a given factor. For instance, while 2940 has two copies of the factor 2 and 3150 has one, I did not take *three* copies of the factor 2. There are only two columns with 2 as their factor, so I took only two copies of that factor when finding the LCM.

Then the LCM is 2 × 2 × 3 × 3 × 5 × 5 × 7 × 7 = 44,100

First, I need to find the prime factorizations:

3)27  2)90  2)84
 ---   ---   ---
 3)9  3)45  2)42
  --   ---   ---
   3  3)15  3)21
       ---   ---
         5     7

Then I will list these factorizations neatly, with each copy of each factor getting its own column:

 27:     3×3×3
 90: 2  ×3×3  ×5
 84: 2×2×3      ×7

Then the GCF (being the product of the shared factors — so its the product of all the full columns) and the LCM (being the product of all factors — so its the product of all of the columns) are given by:

GCF:     3         = 3
---:--------------
 27:     3×3×3
 90: 2  ×3×3  ×5
 84: 2×2×3      ×7
---:--------------
LCM: 2×2×3×3×3×5×7 = 3,780

Then my answer is:

First I factor the numbers and list their prime factorizations:

  3:       3
  6: 2     3
  8: 2 2 2

Then my GCF and LCM are given by:

GCF:         = 1
---:--------
  3:       3
  6: 2     3
  8: 2 2 2
---:--------
LCM: 2 2 2 3 = 24

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, because they have no common factor (other than the "trivial" factor of 1).

Then my answer is:

First I factor the polynomials:

...and:

Then I list these factors out, nice and neat:

The LCM is the product of the entries from all of the columns:

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.