Counting Relatively Prime Integers <500

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In summary, counting relatively prime integers below 500 involves finding the number of positive integers that have no common factors with 500. This can be achieved by using the Euler's totient function, which calculates the number of positive integers less than n that are relatively prime to n. The result shows that there are 200 relatively prime integers below 500, with 1 being the first and 499 being the last. These numbers are important in various mathematical applications, such as cryptography and number theory.
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Caldus
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Having a lot of trouble with this problem as well:

How many integers less than 500 are relatively prime to 500?

I know that when two numbers are relatively prime, that means that the greatest common divisor of those two numbers is 1. But I can't figure out a formula that uses sets in order to calculate this.

If someone could point me in the right direction with this, then it would be greatly appreciated.
 
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  • #2
A number is relatively prime with 500 iff it is not divisibel by 5 or 2. You can count the number that are divisible by two, those divislbe by 5, but you've over counted those divisible by 10. this is the inclusion exclusion principle. if you like probability it is the same as saying P(AuB) = P(A) + P(B) - P(AnB)
 
  • #3
Also, there's a function for this very purpose; the Euler-totient (aka Euler-phi or simply phi) function. The answer you're looking for would then be &phi;(500).

Information on this function can be found at http://mathworld.wolfram.com/TotientFunction.html
 

Related to Counting Relatively Prime Integers <500

1. What are relatively prime integers?

Relatively prime integers are two positive integers that do not share any common factors besides 1. In other words, their greatest common divisor is 1.

2. Why is counting relatively prime integers important?

Counting relatively prime integers is important in number theory and cryptography. It allows us to find the number of possible solutions to certain equations and helps in creating secure encryption algorithms.

3. How do you determine if two numbers are relatively prime?

To determine if two numbers are relatively prime, you can use the Euclidean algorithm to find their greatest common divisor. If the greatest common divisor is 1, then the numbers are relatively prime.

4. What is the significance of the limit of 500 in this problem?

The limit of 500 in counting relatively prime integers is used to limit the range of numbers being considered. This allows for easier calculation and analysis of the numbers.

5. Are there any patterns or formulas that can be used to count relatively prime integers?

Yes, there are several patterns and formulas that can be used to count relatively prime integers. For example, Euler's totient function can be used to calculate the number of relatively prime integers less than a given number. Other patterns and formulas can be found through further research and study in number theory.

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