Finding GCD of x,c,y,z: Fast & Easy

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SUMMARY

The discussion focuses on efficiently calculating the greatest common divisor (GCD) of the expression gcd(C(x, c, y), z), where C(x, y) denotes combinations. The variables x, y, and z are large integers, with x potentially having up to 1000 digits. The participants suggest utilizing prime factorization techniques in conjunction with combinatorial formulas to streamline the GCD computation process.

PREREQUISITES
  • Understanding of GCD and its properties
  • Familiarity with combinatorial mathematics, specifically combinations
  • Knowledge of prime factorization techniques
  • Experience with handling large integers in computational contexts
NEXT STEPS
  • Research efficient algorithms for calculating GCD, such as the Euclidean algorithm
  • Explore combinatorial functions and their applications in number theory
  • Study prime factorization methods for large integers
  • Investigate programming libraries that handle large number computations, such as GMP (GNU Multiple Precision Arithmetic Library)
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Mathematicians, computer scientists, and software developers interested in number theory, particularly those working with large integers and GCD calculations.

smslca
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can we find the value of gcd(x c y , z) easily and very fast using a computer.

where
1. "c" represents "combinations" used in 'permutations and combinations'.
2. x is very very large number (ex: may be of 100 or 1000 numerical digits)
3. y is also large having 2 to 5 digits less than x.
4. z is also large having the same number of digits as x.
 
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What are some ideas that you have?

Not having put much thought into this, I would try to use the formula for C(x,y) to try to determine the prime factorization.
 

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