Relatively Prime Numbers
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:
Comparing two numbers to find out if they're relatively prime
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?
Finding the greatest common factor
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.
Applications and fun facts about relatively prime numbers
These numbers are significant in various areas of mathematics and have interesting properties.
- Fractions in lowest terms: When a fraction is in its simplest form (lowest terms), the numerator and the denominator are relatively prime. This ensures that the fraction cannot be simplified any further.
- Euler's Totient Function: Relatively prime numbers play a crucial role in Euler's totient function, which is used to count the number of positive integers less than or equal to a given number that is relatively prime to that number. This function has essential applications in number theory and cryptography.
- Chinese Remainder Theorem: In number theory, the Chinese Remainder Theorem deals with solving a system of linear congruences where the moduli are pairwise relatively prime. This theorem has applications in computing, cryptography, and algebra.
- LCM and GCD: For any two relatively prime numbers a and b, their least common multiple (LCM) is equal to the product of the numbers, i.e., $\mathrm{LCM}(a,b)=ab$ . This property is a direct result of the fact that their greatest common divisor (GCD) is 1.
Topics related to the Relatively Prime Numbers
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