ALGEBRA EASY FINDING GREATEST COMMON DENOMINATOR
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| A factor of a number is a number that divides evenly into the number. |
The factors of 2 are 1 and 2. The factors of 5 and 5 and 1. 2 and 5 aren't
very interesting from the standpoint of factors, because they are primes.
(See How to Prime Factor a Number for more
about primes.) In terms of factors we can say that a number larger than
1 is a prime if its only factors are 1 and itself. Composites are more interesting
with respect for factors. The factors of 12 are 1, 2, 3, 4, 6, and 12. The
factors of 250 are 1, 2, 5, 10, 25, 50, 125, and 250.
Multiples
| A multiple of a number is a number that you can get by multiplying
it by something. |
So, the multiples of 2 are 2,4,6,8,10,12,..., and the multiples of 5 are
5,10,15,20, 25,.... The multiples are the numbers that you get when you
count by the number.
Factors and Multiples
Factors and multiples are sort of opposites. If one number is a factor of
another number, then the other number is a multiple of it and vice versa.
| 'A is a factor of B' means the same thing as 'B is a multiple of A'. |
For example, 2 is a factor of 12 and 12 is a multiple of 2. 125 is a multiple
of 5 and 5 is a factor of 125. 9 is a factor of 900 and 900 is a multiple
of 9. 70 is a multiple of both 7 and 10, and both 7 and 10 are factors of
70.
It is important to not get confused which one is which. The multiples are
the big ones, and the factors are the little ones. Perhaps it will help
if you remember that the multiple is the result of a multiplication
and the factor isn't. Or associate multiple with multitudes
so that it will just sound big, so that you can remember that it is the
big one, and the factor is the other one, so it must be the little one.
Common
Common in this setting means something that two or more numbers share. For
example a common multiple of 4 and 6 would be a number that is both a multiple
of 4 and a multiple of 6. A common factor of 4 and 6 would be a number that
is a factor of both 4 and 6. 12 is a common multiple of 4 and 6, because 12
is the result of multiplying 3 times 4 and also the result of multiplying
2 times 6. 2 is a common factor of 4 and 6, because 2 divides evenly into
4, and it also divides evenly into 6, 2x2=4 and 2x3=6. It is easy to find
small common factors and big common multiples, because 1 is always a common
factor of any collection of numbers and if you multiply all of the numbers
together you will always get a common multiple. But small common multiples
and large common factors are harder to find.
Relative Primes
| Two numbers are called relative prime if their only common factor
is 1. |
If the numbers themselves are prime they will always be relative prime,
so 3 and 5 are relative prime, but they don't have to be prime to be relative
prime. For example, 8 and 27 are relative prime, but definitely not primes
themselves. Even a number with lots of factors like 60 can be relatively
prime with respect to another number, for example 60 and 121 are relatively
prime. Perhaps you can think of some other interesting examples of relative
primes that are not primes.
Finding GCFs and LCMs by Guessing
Now that we know what factors and multiples and common factors and common
multiples are, we are ready to understand what LCMs and GCFs are. They mean
exactly what their words say. The LCM of a collection of numbers is the least
or smallest number that is a common multiple of them. So if we took all of
the common multiples of the numbers and lined them up and asked which one
was the smallest, then that would be the least common multiple. Similarly
the GCF of a collection of numbers is the greatest or largest number that
is a common factor of the numbers. If we took all of the common factors of
those numbers and lined them up and then took that largest one, that would
be the greatest common factor. As I mentioned earlier, it is easy to find
large common multiples and small common factors, so this is the more interesting
task, finding a common factor that isn't any smaller than necessary and a
common multiple that isn't any larger than necessary. For many collections
of numbers, particularly when they aren't too large, you can do this by guessing
and playing around with the numbers.
To find GCFs this way, if you don't see a large common factor right away,
try looking for primes that divide the numbers evenly, and then if you find
more than one of them try multiplying them together and see if that is still
a common factor. For example if you have 12 and 18, you could see that both
2 and 3 are common factors, so maybe 6 will also be a common factor, and
it is. Is there a larger one? Not too likely because 6 is a pretty large
factor just for 12 alone, and in fact the only larger factor that 12 alone
has is 12, which is clearly not a factor of 18.
For LCMs there are a couple of techniques you can use. You can always find
a common multiple by multiplying the numbers, so for example 15 is a common
multiple of 3 and 5, because it is 5x3 and 3x5. 12x18=216, so 216 is a common
multiple of 12 and 18, since it is 18x12 and 12x18. If you are dealing with
only two numbers, this will be the smallest one, so the LCM, whenever the
two numbers are relatively prime, but if the numbers have a common factor
larger than 1, there should be a smaller number that will work. 15 is the
LCM of 3 and 5, but 216 is not the LCM of 12 and 18, but at least you know
from this that the LCM for them can be no larger than 216.
On the other end of the scale, if one of your numbers is a multiple of
the other, then it is easy to find the LCM, it is just the bigger number.
So the LCM of 8 and 4 is 8, and the LCM of 5 and 15 is 15. For numbers like
12 and 18, you might just think of it without even knowing how you do it.
Often times when I ask someone for a common multiple of two numbers without
even specifying that I want the LCM, they tend to instinctively give me
the LCM, so if you can do it that way, that's fine. Think of a common multiple
of 12 and 18, and perhaps 36 just comes to mind, and then you make sure
you can't think of a smaller one, and since 18 is the only smaller multiple
of just 18, and it is not a multiple of 12, it must be the answer.
But if that doesn't work for you, there is a more systematic way you can
go about the task in a problem like this. What you can do is you can take
the largest of the numbers, and start listing successive multiples of it,
by multiplying it by 2, 3, 4, etc., until you come up with one that is a
multiple of the other numbers. So in this case it would work in just one
step, since 2x18=36, and 36 is indeed a multiple of 12, but if that didn't
work you could go on to 3x18, 4x18, until one of them was a multiple of
12. The reason to use the largest number when doing this is that if you
do that, you won't have to go as far. In this case if you used 12 instead,
you would say 2x12=24, no 24 is not a multiple of 18, 3x12=36, 36 is, so
that works, and you still get the right answer but it takes one more step.
For another example, let's say we want to find the LCM of 50 and 60. Write
down successive multiples of 60 until you get one that is also a multiple
of 50. 60x2=120, no, 60x3=180, no, 60x4=240, no, 60x5=300, yes, we've got
it, so 300 is the LCM of 50 and 60. Notice if you simply multiplied the
two numbers together you would get 3000, which is much bigger. That is because
50 and 60 have a common factor of 10. Now here is something interesting.
If you divide 3000 by the common factor 10, you get 300, the LCM. Actually
for two numbers, this is a trick that will always work, and it might be
useful in a case like this where it is easy to see what the GCF is. 10 is
not just any old common factor of 50 and 60, it is the greatest common factor,
and it turns out that if you divide the product by the GCF, that will always
give you the LCM. If you want to use that trick, use it, but do realize
that it is only a good trick for problems where the numbers are easy to
multiply and the GCF is easy to find. For 12 and 18, you could do it too,
12x18=216, 216/6=36, but here is it probably easier to just multiply 18
by 2.
For finding LCMs and GCFs of larger numbers and more than 2 numbers there
is a more systematic method that is also important to learn for when you
later learn to do this sort of thing with algebraic expressions. This method
involves prime factoring. To learn how to prime factor numbers, see my article
How to Prime Factor a Number. But since too
many people learn these method without understanding them, before I talk
about them I want to talk a little about the relationships between the prime
factorization of factors and multiples.
Factors and Multiples in Prime Factored Form
Here's the game. Let's say we have our numbers in prime factored form and
we want to tell whether one number is a multiple or a factor of the other.
How can we do it? Is there a good way to tell this from the prime factored
form? For example, here are two numbers prime factored.
24x3x5, 3x5x7
Without multiplying out to see what the numbers are, can we tell if the
first is a multiple or a factor of the second? How about these two
numbers?
3x5, 3x5x7
For these two we can see something interesting. The factorization of the
first one is contained in that of the second one, which makes the first
one a factor of the second one and the second one a multiple of the first
one. 3x5 divides 3x5x7 evenly, namely when you divide 3x5x7 by 3x5, you
get 7, and since 3x5x7=(3x5)x7, 3x5x7 can be obtained by multiplying something
by 3x5, namely 7, so it is a multiple of it. In general this test will always
work.
| In prime factored form one number is a multiple of another number
if its factorization contains the factorization of the other number.
In prime factored form one number is a factor of another number if
its factorization is contained in the factorization of the other number.
|
