Finding the Remainder when 77777^8 is Divided by 9

Understanding modular arithmetic is a fundamental skill in number theory and can be particularly useful for solving problems related to remainders. In this guide, we will walk through the steps to find the remainder when 77777^8 is divided by 9. We will use several mathematical theorems and properties to simplify the calculation.

1. Introduction to the Problem

We are tasked with determining the remainder when 7^7777^8 is divided by 9. This problem can be approached using Euler's Theorem and properties of modular arithmetic.

2. Using Modular Arithmetic and Euler's Theorem

Euler's Theorem states that for any integer a and n that are coprime, the following holds:

aφ(n) ≡ 1 (mod n)

Here, φ(n) is Euler's totient function, which counts the number of integers up to n that are coprime with n.

Step 1: Determine the Totient Function of 9

The number 9 can be factored as 9 3^2. Therefore, the totient function φ(9) is calculated as:

φ(9) 9(1 - 1/3) 6

Since 7 is coprime with 9, Euler's Theorem can be applied here:

7^6 ≡ 1 (mod 9)

Step 2: Simplify the Exponent

We need to find the value of the exponent 7777^8 modulo 6. Let's calculate:

7777 ≡ 1 (mod 6)

Therefore:

7777^8 ≡ 1^8 ≡ 1 (mod 6)

Let t 7777^8. We can write:

7^7777^8 7^t

Since t ≡ 1 (mod 6), we have:

7^t ≡ 7^1 ≡ 7 (mod 9)

3. Verification Using the Property of Modular Arithmetic

We can also verify this using the property of modular arithmetic:

If a ≡ b (mod φ(n)) and gcd(a, n) 1, then:

a^d ≡ a^b (mod n)

Here, 7777 ≡ 1 (mod 6), and since gcd(7, 9) 1, we have:

7777^8 ≡ 1^8 ≡ 1 (mod 6)

Thus:

7^{7777^8} ≡ 7^1 ≡ 7 (mod 9)

4. Conclusion

The remainder when 7^{7777^8} is divided by 9 is 7. This result can be confirmed by both applying Euler's Theorem and using properties of modular arithmetic.

Key Learning Points:

Understanding how to apply Euler's Theorem in modular arithmetic. Using the property of modular arithmetic to simplify calculations. Finding remainders when raised exponents are involved.

This method is widely applicable and can be used to solve similar problems involving large exponents and modular arithmetic.