As I stated in my earlier post on factoring, I've gotten confused before by factors and multiples. We went over factoring in that post, and now we are going to review multiples.
A multiple is the opposite of a factor. Whereas factoring uses division, multiples use multiplication.
Essentially, a multiple is the result of one number being multiplied by another number which increases the first number.
In factoring, the amount of numbers we can generate are finite because we are breaking an existing number down. With multiples, the numbers generated are infinite because we are increasing the existing number. Except for 0, of course because zero increased by any number will always be 0.
On that note, 0 broken down by any number will always be 0 as well. Any number over 0, however, is infinite. Unless it is 0 over 0 which is undefined. Just an interesting tidbit for you.
Anyway...let's find the multiples of 5
5*1 = 5
5*2 = 10
5*3 = 15
5*4 = 20
5*5 = 25
...
5*100 = 500
...
5*1000 = 5000
You see how this can go on forever, right?
This is why we don't solve for the greatest common multiple like we do for the greatest common factor. It could take a while.
Finding the least common multiple is actually very similar to finding the greatest common factor.
The least common multiple is the lowest number in a set of numbers that is a multiple of all the numbers in that set. It's finding the smallest common number that they both share. Easy, right?
So let's try and find the least common multiples of 2 and 50:
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50
Multiples of 50: 50, 100, 150, 200
The least common multiple of 2 and 50 is 50.
Woah, we could have started with the higher number, 50, and not had to write out so many multiples for 2! Basically, you're better off to start with the greater multiple first.
This can be quite time consuming though. Look at how many multiples of 2 we had to list before we got to a multiple of 50 that they both shared.
There's something easier that we can do. We can piggy back on what we learned about prime factorization - decomposing a number down to its prime numbers (sounds so morbid, doesn't it?) This is very useful when we have multiple sets of numbers.
In my post about factoring, I compared prime factorization to breaking down a molecule to its atoms. Now, we are breaking several molecules down to their atoms and using those atoms to create the smallest molecule that we can use to recreate the original molecules at least once.
Now let's find the least common multiples of 20, 50, and 11
11 is already a prime number so it equals 1*11
20 50
/ \ / \
2 10 5 10
/ \ / \
2 5 2 5
20 = 2*2*5
50 = 5*2*5
Looking at the multiples as a group, we look for the most repeated prime numbers in each multiple (if one of the decompositions contains one number that the others do not, you underline that number too):
11 = 1*11
20 = 2*2*5
50 = 5*2*5
We see that the prime numbers in this group of multiples are 1, 2, 5, and 11.
For each number, figure out which prime numbers occur the most and use those numbers (I highlighted them for you).
Now we take those numbers and multiply them:
1*2*2*5*5*11 = 1100
1100 is the least common multiple of 11, 20, and 50.
Our goal when finding the least common multiple is to find the smallest number that the other numbers share when they are increased through multiplication. Otherwise we don't have the least common multiple, we have just a common multiple. D'oy. You could find a common multiple by just multiplying 11*20*50, but the product would be 11000, a far larger number than 1100.
I am wondering, why do I need to find the most repeated prime numbers in each multiple? Wouldn't we find the least common multiple by just taking each multiple in the group and multiplying that together?
Let's try it out.
1*2*5*11 = 110
110/11 = 10
110/20 = 5.5
110/50 = 2.2
Doesn't work. Boo.
Generate some problems to help you cement your knowledge of mutliples!
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