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You've heard of prime numbers.. But what about "relatively prime" numbers? Yes, these numbers are real, and understanding how they work will help us solve a wide range of math problems. Let's discover more about relatively prime numbers and learn why they're so important:
We know that prime numbers are numbers that can only be divided by themselves and one. For example, 2, 3, 5, 7, and 11 are all prime numbers.
But when we say that numbers are "relatively prime," we are talking about the relationship between two numbers rather than describing the properties of one number.
For numbers to be relatively prime, their greatest common factor must be 1.
So how does that work?
In order to find out whether two numbers are relatively prime, we need to find their greatest common factor -- often abbreviated as "GCF."
You might recall that the greatest common factor of two numbers is the largest number that is a factor of both. For example, 60 and 42 share factors 2, 3 and 6.
Now let's use that principle to find out whether two numbers are relatively prime.
Consider these two numbers:
20 and 33
20 has factors 1, 2, 4, 5, 10, and 20
33 has factors 1, 3, and 11
Let's scan all of these factors together to find which ones are common. Very quickly, we can see that there is only one common factor -- 1.
So are these two numbers relatively prime? Yes, because their GCF is 1.
We can use the same principles with more complicated problems -- such as polynomials.
Consider these two polynomials:
$3{x}^{2}+21x+18$ and $5x+10$
Even though these two numbers might seem complicated, all we really need to do is look at the coefficients.
$3{x}^{2}+21x+18$ can be rewritten as $3(x+1)(x+6)$
$5x+10$ can be rewritten as $5(x+2)$
Because none of the factors are the same across both sets of polynomials we can be sure they are relatively prime.
If these two polynomials shared a binomial factor like $\left(x+1\right)$ , we would know right away that they were not relatively prime.
These numbers are significant in various areas of mathematics and have interesting properties.
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