Number of positive divisiors with gcd condition

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SUMMARY

The discussion centers on finding a general formula for the number of positive divisors \( b \) of a positive integer \( n \) such that the greatest common divisor \( (a, b) = 1 \), where \( a \) is a positive divisor of \( n \). It is established that while there is no direct formula for this specific case, the well-known divisor function \( d(n) = \prod (r_i + 1) \) can be adapted. The adapted formula excludes the primes that divide \( a \) from the product used to calculate \( d(n) \).

PREREQUISITES
  • Understanding of the divisor function \( d(n) \)
  • Knowledge of prime factorization of integers
  • Familiarity with greatest common divisor (GCD) concepts
  • Basic number theory principles
NEXT STEPS
  • Research the divisor function \( d(n) \) in detail
  • Study prime factorization techniques and their applications
  • Learn about the properties of the greatest common divisor (GCD)
  • Explore related number theory topics on coprime integers
USEFUL FOR

Mathematicians, number theorists, and students studying divisor functions and GCD properties will benefit from this discussion.

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Let n be a positive integer and a be a positive divisor of n. Is there any general formula to find the number of positive divisors b of n such that (a,b)=1 ?.
 
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No. That would be a formula to factor a number but there isn't any.
 
Yes. There is a general formula. First, you should understand the well known formula d(n)=∏(ri+1) for the number of divisors of n, where n=∏piri is the prime factorization of n. See for example wikipedia or OEIS.

In your case, it is the same product as for d(n), except you don't run it over all primes, but exclude the primes dividing a.
 

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