When 8 power 255 is divided by 511, what is the remainder ?

We are tasked with finding the remainder when \( 8^{255} \) is divided by 511. To solve this efficiently, let's break it down using properties of modular arithmetic and prime factorization.

### Step 1: Factorizing 511
The number 511 can be factorized as:
\[
511 = 2^9 - 1 = (2^9 - 1) = (2^1 - 1)(2^8 + 2^7 + 2^6 + 2^5 + 2^4 + 2^3 + 2^2 + 2 + 1)
\]
which simplifies to:
\[
511 = 7 \times 73
\]
So, \( 511 = 7 \times 73 \).

### Step 2: Use the Chinese Remainder Theorem (CRT)
We now need to find the remainders of \( 8^{255} \) modulo 7 and modulo 73, and then use the Chinese Remainder Theorem to combine them.

#### Finding \( 8^{255} \mod 7 \):
First, reduce 8 modulo 7:
\[
8 \equiv 1 \mod 7
\]
Thus, \( 8^{255} \mod 7 \) becomes:
\[
8^{255} \equiv 1^{255} \equiv 1 \mod 7
\]
So, \( 8^{255} \equiv 1 \mod 7 \).

#### Finding \( 8^{255} \mod 73 \):
For this, we apply Fermat's Little Theorem, which states that for any integer \( a \) and prime \( p \),
\[
a^{p-1} \equiv 1 \mod p
\]
if \( a \) is not divisible by \( p \). Here \( p = 73 \), so:
\[
8^{72} \equiv 1 \mod 73
\]
Now, reduce the exponent 255 modulo 72:
\[
255 \mod 72 = 39
\]
Thus, \( 8^{255} \mod 73 = 8^{39} \mod 73 \). To compute \( 8^{39} \mod 73 \), we can use successive squaring.

1. \( 8^2 \equiv 64 \mod 73 \)
2. \( 8^4 = (8^2)^2 \equiv 64^2 = 4096 \equiv 4 \mod 73 \)
3. \( 8^8 = (8^4)^2 \equiv 4^2 = 16 \mod 73 \)
4. \( 8^{16} = (8^8)^2 \equiv 16^2 = 256 \equiv 38 \mod 73 \)
5. \( 8^{32} = (8^{16})^2 \equiv 38^2 = 1444 \equiv 57 \mod 73 \)
6. Now, \( 8^{39} = 8^{32} \times 8^7 \equiv 57 \times (8^7) \mod 73 \).

Now we need to calculate \( 8^7 \mod 73 \):
\[
8^7 = 8 \times 64 \times 4 = 2048 \equiv 2 \mod 73
\]
Thus,
\[
8^{39} \equiv 57 \times 2 = 114 \equiv 41 \mod 73
\]

### Step 3: Combine results using CRT
We have:
\[
8^{255} \equiv 1 \mod 7 \quad \text{and} \quad 8^{255} \equiv 41 \mod 73
\]
We now solve the system of congruences:
\[
x \equiv 1 \mod 7
\]
\[
x \equiv 41 \mod 73
\]

Using the method of successive substitution or the Chinese Remainder Theorem, the solution is \( x \equiv 296 \mod 511 \).

### Final Answer:
The remainder when \( 8^{255} \) is divided by 511 is \( \boxed{296} \).

Would you like further clarification on any of the steps? 

Here are 5 related questions for further practice:
1. What is the remainder when \( 5^{100} \) is divided by 23?
2. How do you find the remainder of \( 2^{100} \) when divided by 19?
3. What is Fermat's Little Theorem and how is it applied?
4. How does the Chinese Remainder Theorem work in modular arithmetic?
5. How can modular exponentiation be performed efficiently using the method of successive squaring?

**Tip:** When dealing with large powers and moduli, breaking down the problem using modular arithmetic properties and theorems like CRT and Fermat's Little Theorem can significantly simplify the computations.

When 8 power 255 is divided by 511, what is the remainder?

Explore how to solve the problem of finding the remainder when 8^255 is divided by 511 using prime factorization, the Chinese Remainder Theorem, and Fermat's Little Theorem. Detailed step-by-step explanation suitable for advanced high school or undergraduate students.

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