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We are asked to find the least residue of \( 67^{821} \mod 821 \) using **Fermat's Little Theorem**. Here's the solution process:

### Fermat's Little Theorem:
Fermat’s Little Theorem states that if \( p \) is a prime number and \( a \) is an integer not divisible by \( p \), then:
\[
a^{p-1} \equiv 1 \pmod{p}
\]
For this problem:
- \( a = 67 \)
- \( p = 821 \) (since 821 is prime)

### Step 1: Simplify the Exponent using Fermat’s Little Theorem
By Fermat's Little Theorem, \( 67^{820} \equiv 1 \pmod{821} \) because 821 is prime.

We want to find \( 67^{821} \mod 821 \). This can be written as:
\[
67^{821} = 67^{820} \times 67
\]
Using Fermat's Little Theorem:
\[
67^{820} \equiv 1 \pmod{821}
\]
Thus:
\[
67^{821} \equiv 67 \pmod{821}
\]

### Step 2: Conclusion
The least residue of \( 67^{821} \mod 821 \) is:
\[
67
\]

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 \( a \) were divisible by \( p \) 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.

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

Learn how to find the least residue of 67^821 modulo 821 using Fermat's Little Theorem. This problem involves modular arithmetic and number theory concepts, suitable for students in Grades 11-12.

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