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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 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:
\[
a^{p-1} \equiv 1 \pmod{p}
\]
Here, \( p = 11 \) and \( a = 97 \).

First, we reduce 97 modulo 11:
\[
97 \mod 11 = 9
\]
So, we need to find the remainder when \( 9^{97^{97}} \) is divided by 11.

By Fermat's Little Theorem, since \( 9^{10} \equiv 1 \pmod{11} \), we can reduce the exponent \( 97^{97} \) modulo 10.

Now, let's reduce \( 97 \) modulo 10:
\[
97 \mod 10 = 7
\]
So \( 97^{97} \equiv 7^{97} \pmod{10} \).

Since \( 7^4 \equiv 1 \pmod{10} \) (because \( 7^4 = 2401 \equiv 1 \mod 10 \)), we can reduce \( 97 \) modulo 4:
\[
97 \mod 4 = 1
\]
Thus:
\[
7^{97} \equiv 7^1 \equiv 7 \pmod{10}
\]
Therefore:
\[
97^{97} \equiv 7 \pmod{10}
\]

Now, the problem reduces to finding:
\[
9^7 \mod 11
\]

Let's calculate \( 9^7 \mod 11 \):
\[
9^2 = 81 \equiv 4 \pmod{11}
\]
\[
9^4 = (9^2)^2 = 16 \equiv 5 \pmod{11}
\]
\[
9^7 = 9^4 \times 9^2 \times 9 = 5 \times 4 \times 9 = 180 \equiv 4 \pmod{11}
\]

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. How does Fermat's Little Theorem apply to modular arithmetic problems?
2. What are the conditions under which Fermat's Little Theorem can be used?
3. Can Fermat's Little Theorem be applied if the base is divisible by the modulus?
4. How do exponentiation cycles help in simplifying large powers in modular arithmetic?
5. What are other theorems useful for modular arithmetic problems?

**Tip:** When working with large exponents in modular arithmetic, always look for patterns or cyclic properties to simplify calculations.

Discover how to find the remainder of 97 raised to the power of 97 to the power of 97 modulo 11 using Fermat's Little Theorem. Step-by-step solution with detailed explanations and calculations included.

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