Math Problem Statement

Using Fermat's little theorem, find the least residue of 67^821 modulo 821. (Note that 821 is prime.)

Solution

We are asked to find the least residue of 67821mod82167^{821} \mod 821 using Fermat's Little Theorem. Here's the solution process:

Fermat's Little Theorem:

Fermat’s Little Theorem states that if pp is a prime number and aa is an integer not divisible by pp, then: ap11(modp)a^{p-1} \equiv 1 \pmod{p} For this problem:

  • a=67a = 67
  • p=821p = 821 (since 821 is prime)

Step 1: Simplify the Exponent using Fermat’s Little Theorem

By Fermat's Little Theorem, 678201(mod821)67^{820} \equiv 1 \pmod{821} because 821 is prime.

We want to find 67821mod82167^{821} \mod 821. This can be written as: 67821=67820×6767^{821} = 67^{820} \times 67 Using Fermat's Little Theorem: 678201(mod821)67^{820} \equiv 1 \pmod{821} Thus: 6782167(mod821)67^{821} \equiv 67 \pmod{821}

Step 2: Conclusion

The least residue of 67821mod82167^{821} \mod 821 is: 6767

Would you like more details on Fermat’s Little Theorem or the steps involved?

Here are 5 related questions to expand your understanding:

  1. How is Fermat's Little Theorem used for solving modular exponentiation problems?
  2. What would happen if the base aa were divisible by pp in Fermat's Little Theorem?
  3. Can Fermat's Little Theorem be used for composite moduli, and if so, how is it adapted?
  4. What are some applications of Fermat’s Little Theorem in cryptography?
  5. 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