Math Problem Statement
calculate 5^-1 mod 23 by fermats theorem
Solution
To calculate using Fermat's Little Theorem, we proceed as follows:
Fermat's Little Theorem
Fermat's Little Theorem states that if is a prime number and is any integer such that , then: This implies that: Thus, the modular inverse of is .
Applying Fermat's Theorem
Here, and (a prime number). To compute , we need to calculate:
Now, let's compute .
We can use modular exponentiation to simplify the process:
Now, combine powers:
Thus, .
Final Answer
Would you like further details on modular exponentiation or Fermat's theorem?
Here are 5 related questions:
- How can Fermat's Little Theorem be used to calculate larger modular inverses?
- What are other ways to calculate the modular inverse aside from Fermat's theorem?
- Can the extended Euclidean algorithm be used to find the inverse of 5 modulo 23?
- How do we compute large powers mod efficiently using exponentiation by squaring?
- What is the relationship between Fermat's Little Theorem and Euler's Theorem?
Tip: To compute modular inverses efficiently for larger numbers, consider using the extended Euclidean algorithm, which is faster for practical use than Fermat's theorem.
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Math Problem Analysis
Mathematical Concepts
Modular Arithmetic
Modular Inverses
Number Theory
Formulas
a^(p-2) ≡ a^(-1) mod p
Modular Exponentiation
Theorems
Fermat's Little Theorem
Suitable Grade Level
Grades 11-12
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