57^(th) digit in decimal expansion

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SUMMARY

The discussion focuses on calculating the 57th digit in the decimal expansion of the fraction $\frac{1}{57}$. It emphasizes the use of modular arithmetic and Euler's phi function to determine the period of the decimal expansion. By leveraging these mathematical tools, one can compute the desired digit without needing to calculate all preceding digits in the expansion. This approach streamlines the process and enhances efficiency in finding specific digits in repeating decimals.

PREREQUISITES
  • Understanding of modular arithmetic
  • Familiarity with Euler's phi function
  • Knowledge of decimal expansions and their periodicity
  • Basic skills in number theory
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  • Study modular arithmetic applications in number theory
  • Learn about Euler's phi function and its significance
  • Explore methods for calculating decimal expansions of fractions
  • Investigate periodicity in decimal representations
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Mathematicians, students of number theory, and anyone interested in advanced calculations involving decimal expansions and modular arithmetic.

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Calculation of $57^{th}$ digit in the decimal expansion of $\displaystyle \frac{1}{57}$
 
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jacks said:
Calculation of $57^{th}$ digit in the decimal expansion of $\displaystyle \frac{1}{57}$

If you are allowed mod calculation and Euler's phi function, you can use >>this<<.
It gives you the period of the decimal expansion, meaning it suffices to calculate a decimal that comes much earlier in the expansion.
 

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