Find the remainder when 97 97 ^ 97 is divided by 11

To find the remainder when \( 97^{97^{97}} \) is divided by 11, we can break the problem down using modular arithmetic and Fermat's Little Theorem.

### Step 1: Reduce the base modulo 11
We first reduce 97 modulo 11:
\[
97 \mod 11 = 9
\]
So, we need to find the remainder of \( 9^{97^{97}} \) when divided by 11.

### Step 2: Apply 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:
\[
a^{p-1} \equiv 1 \pmod{p}
\]
Here, \( p = 11 \), so:
\[
9^{10} \equiv 1 \pmod{11}
\]

### Step 3: Simplify the exponent using modulo 10
Since \( 9^{10} \equiv 1 \pmod{11} \), we need to simplify the exponent \( 97^{97} \) modulo 10:
\[
97 \mod 10 = 7
\]
So we now need to compute:
\[
7^{97} \mod 10
\]

### Step 4: Simplify the exponent modulo 4
Notice that \( 7^4 \equiv 1 \pmod{10} \). Therefore, we can reduce \( 97 \) modulo 4:
\[
97 \mod 4 = 1
\]
Thus:
\[
7^{97} \equiv 7^1 \equiv 7 \pmod{10}
\]

### Step 5: Compute \( 9^7 \) modulo 11
Now, we need to compute \( 9^7 \mod 11 \):
\[
9^2 = 81 \equiv 4 \pmod{11}
\]
\[
9^4 = (9^2)^2 = 16 \equiv 5 \pmod{11}
\]
\[
9^6 = 9^4 \times 9^2 = 5 \times 4 = 20 \equiv 9 \pmod{11}
\]
\[
9^7 = 9^6 \times 9 \equiv 9 \times 9 = 81 \equiv 4 \pmod{11}
\]

So, the remainder when \( 97^{97^{97}} \) is divided by 11 is \( \boxed{4} \).

Would you like more details or have any questions?

Here are some related questions:

1. Why is Fermat's Little Theorem useful in modular arithmetic problems?
2. How can you compute large exponents modulo small numbers efficiently?
3. What is the significance of the exponent's remainder when simplifying problems using modular arithmetic?
4. How do cyclic patterns in powers help simplify modular arithmetic problems?
5. Can the method used here be applied to different bases and moduli?

**Tip:** When dealing with large exponents, reducing the base and the exponent modulo smaller numbers helps simplify the calculation process.

Learn how to find the remainder when 97 raised to the power of 97^97 is divided by 11 using modular arithmetic and Fermat's Little Theorem. Step-by-step solution with detailed explanations.

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