Counting Relatively Prime Integers <500

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SUMMARY

The discussion focuses on calculating the number of integers less than 500 that are relatively prime to 500. It is established that a number is relatively prime to 500 if it is not divisible by 2 or 5. The inclusion-exclusion principle is applied to avoid over-counting integers divisible by both 2 and 5. The Euler-totient function, denoted as φ(500), is identified as the key tool for determining the count of these integers, providing a direct method for the calculation.

PREREQUISITES
  • Understanding of the concept of relatively prime integers
  • Familiarity with the inclusion-exclusion principle
  • Knowledge of the Euler-totient function
  • Basic number theory concepts
NEXT STEPS
  • Study the properties and applications of the Euler-totient function
  • Learn how to apply the inclusion-exclusion principle in combinatorial problems
  • Explore advanced number theory topics related to prime factorization
  • Investigate the relationship between probability and number theory
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Mathematicians, students of number theory, and anyone interested in combinatorial mathematics or the properties of integers.

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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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)
 
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
 

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