Question 115030: I am struggle with these problems and I need help and explanation on how to do it in the future.
a) Write the prime factorization of 260.
b) Find the greatest common factor (GCF) of 84 and 140.
c) Find the least common multiple (LCM) of 18, 20, 30.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website! To do a prime factorization start dividing by the primes in ascending order.
 , 1 factor of 2
 , 2 factors of 2
 , not an integer, so no more factors of 2
 , not an integer, so no factors of 3
 , 1 factor of 5, and the result is prime so you are done.
The prime factorization is then 
In general, when you are performing this process and you don't find any factors, you can quit trying when you are using the largest prime that is less than the square root of the number you are testing.
The greatest common factor of two numbers is the largest number that divides both of the two numbers evenly. To find a GCF, perform a prime factorization of each of the original numbers.
 , 2
 , 2
 , not an integer, so no more factors of 2 (I usually don't write out this step, because I can see the result is odd and will not be divisible by 2 any further)
 , hence 
 , 2
 , 2
 , not an integer
 , hence 
Now examine the prime factors. The prime factors 84 and 140 have in common are 2, 2, and 7.  and that is your GCF.
For the least common multiple, you are looking for the smallest number that is evenly divisible by your numbers. If you just multiplied 18 * 20 * 30, you would get a common multiple, but it wouldn't necessarily be the smallest. Again, we fall back on the prime factorization process.
 , so 
 , so 
 , so 
Again, we examine the sets of factors, but this time we need to account for the most instances of a factor. In the factorization of 18, there are two 3s and at most only one 3 in any of the others, so our LCM will need two 3s, in the factorization of 20, there are two 2s and at most only one 2 in any of the others, so our LCM will also need two 2s, and we have at most one 5 in any of the factorizations, so the LCM will need one 5. Since there are no other prime factors represented in any of the prime factorizations, we can now say that the LCM is  . You can check that 180 is truly a common multiple by noting that 180 is evenly divisible by 18, 20, and 30. You may notice that this is considerably (60 times) smaller than just multiplying the three numbers together 
Hope this helps,
John
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