Math Problem Statement

calculate 5^-1 mod 23 by fermats theorem

Solution

To calculate 51mod235^{-1} \mod 23 using Fermat's Little Theorem, we proceed as follows:

Fermat's Little Theorem

Fermat's Little Theorem states that if pp is a prime number and aa is any integer such that gcd(a,p)=1\gcd(a, p) = 1, then: ap11modpa^{p-1} \equiv 1 \mod p This implies that: ap2a1modpa^{p-2} \equiv a^{-1} \mod p Thus, the modular inverse of amodpa \mod p is ap2modpa^{p-2} \mod p.

Applying Fermat's Theorem

Here, a=5a = 5 and p=23p = 23 (a prime number). To compute 51mod235^{-1} \mod 23, we need to calculate: 5232mod23=521mod235^{23-2} \mod 23 = 5^{21} \mod 23

Now, let's compute 521mod235^{21} \mod 23.

We can use modular exponentiation to simplify the process:

52=252mod235^2 = 25 \equiv 2 \mod 23 54=(52)2=22=4mod235^4 = (5^2)^2 = 2^2 = 4 \mod 23 58=(54)2=42=16mod235^8 = (5^4)^2 = 4^2 = 16 \mod 23 516=(58)2=162=2563mod235^{16} = (5^8)^2 = 16^2 = 256 \equiv 3 \mod 23

Now, combine powers: 521=516×54×5=3×4×5=6014mod235^{21} = 5^{16} \times 5^4 \times 5 = 3 \times 4 \times 5 = 60 \equiv 14 \mod 23

Thus, 51mod23=145^{-1} \mod 23 = 14.

Final Answer

51mod23=145^{-1} \mod 23 = 14

Would you like further details on modular exponentiation or Fermat's theorem?


Here are 5 related questions:

  1. How can Fermat's Little Theorem be used to calculate larger modular inverses?
  2. What are other ways to calculate the modular inverse aside from Fermat's theorem?
  3. Can the extended Euclidean algorithm be used to find the inverse of 5 modulo 23?
  4. How do we compute large powers mod pp efficiently using exponentiation by squaring?
  5. 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