To calculate $8^{456} \mod 97$, we can use modular exponentiation, which is an efficient way to compute large powers modulo some number. Here's a step-by-step breakdown of how we can solve this:

Step 1: Use 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: $a^{p-1} \equiv 1 \mod p$ In this case, $p = 97$ and $a = 8$, and since 97 is prime, Fermat's Little Theorem tells us that: $8^{96} \equiv 1 \mod 97$

Step 2: Reduce the Exponent Modulo 96

Since $8^{96} \equiv 1 \mod 97$, we can reduce the exponent $456 \mod 96$. This gives us: $456 \div 96 = 4 \text{ remainder } 72$ So: $8^{456} \equiv 8^{72} \mod 97$

Step 3: Compute $8^{72} \mod 97$

Now, we compute $8^{72} \mod 97$ using exponentiation by squaring. We break it down as follows: $8^{72} = (8^2)^{36}$ First, calculate $8^2 \mod 97$: $8^2 = 64 \quad \text{so} \quad 8^2 \mod 97 = 64$ Then, we continue squaring: $64^2 = 4096 \quad \text{and} \quad 4096 \mod 97 = 4096 - (97 \times 42) = 4096 - 4074 = 22$ Now, $8^4 \equiv 22 \mod 97$.

Next, we square again: $22^2 = 484 \quad \text{and} \quad 484 \mod 97 = 484 - (97 \times 5) = 484 - 485 = 99 \mod 97 = 2$ Now, $8^8 \equiv 2 \mod 97$.

Continue squaring: $2^2 = 4 \quad \text{so} \quad 8^{16} \equiv 4 \mod 97$ Next: $4^2 = 16 \quad \text{so} \quad 8^{32} \equiv 16 \mod 97$ And: $16^2 = 256 \quad \text{and} \quad 256 \mod 97 = 256 - (97 \times 2) = 256 - 194 = 62$ Now, $8^{64} \equiv 62 \mod 97$.

Finally: $8^{72} = 8^{64} \times 8^8 \equiv 62 \times 2 = 124 \quad \text{and} \quad 124 \mod 97 = 124 - 97 = 27$

Final Answer:

$8^{456} \mod 97 = 27$

Would you like further details or clarifications on any part of the process? Here are some related questions that might interest you:

1. How does modular exponentiation by squaring work in more general cases?
2. What are other applications of Fermat's Little Theorem?
3. Can we compute large powers mod composite numbers (non-prime moduli)?
4. How do we use Python or another tool to compute modular exponentiation efficiently?
5. What are the properties of powers of numbers in modular arithmetic?

Tip: When dealing with modular exponentiation, breaking down large exponents using Fermat's Little Theorem or Euler’s Theorem can make the problem significantly easier.