Math Problem Statement
Solution
We are asked to find the least residue of using Fermat's Little Theorem. Here's the solution process:
Fermat's Little Theorem:
Fermat’s Little Theorem states that if is a prime number and is an integer not divisible by , then: For this problem:
- (since 821 is prime)
Step 1: Simplify the Exponent using Fermat’s Little Theorem
By Fermat's Little Theorem, because 821 is prime.
We want to find . This can be written as: Using Fermat's Little Theorem: Thus:
Step 2: Conclusion
The least residue of is:
Would you like more details on Fermat’s Little Theorem or the steps involved?
Here are 5 related questions to expand your understanding:
- How is Fermat's Little Theorem used for solving modular exponentiation problems?
- What would happen if the base were divisible by in Fermat's Little Theorem?
- Can Fermat's Little Theorem be used for composite moduli, and if so, how is it adapted?
- What are some applications of Fermat’s Little Theorem in cryptography?
- How can Euler's Theorem generalize Fermat's Little Theorem for non-prime moduli?
Tip: Modular exponentiation can be efficiently computed using the method of exponentiation by squaring to handle large powers and moduli.
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Math Problem Analysis
Mathematical Concepts
Number Theory
Modular Arithmetic
Formulas
a^(p-1) ≡ 1 (mod p) for a not divisible by prime p
Theorems
Fermat's Little Theorem
Suitable Grade Level
Grades 11-12
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