Find remainder when `4444^(4444)` is divided by 9

Answer

To find the remainder when \( 4444^{4444} \) is divided by 9, we can simplify the problem using modular arithmetic. Here’s a step-by-step solution:

### Step 1: Simplify the base modulo 9
First, we need to find \( 4444 \mod 9 \).

To do this, we can sum the digits of 4444:
\[ 4 + 4 + 4 + 4 = 16 \]
Now, we sum the digits of 16:
\[ 1 + 6 = 7 \]
Thus, \( 4444 \equiv 7 \mod 9 \).

### Step 2: Rewrite the expression
Now, we can rewrite the original expression:
\[ 4444^{4444} \equiv 7^{4444} \mod 9 \]

### Step 3: Find the pattern in powers of 7 modulo 9
Next, we will find the powers of 7 modulo 9 to see if there is a repeating pattern:
- \( 7^1 \mod 9 = 7 \)
- \( 7^2 \mod 9 = 49 \mod 9 = 4 \)
- \( 7^3 \mod 9 = 7 \times 7^2 = 7 \times 4 = 28 \mod 9 = 1 \)
- \( 7^4 \mod 9 = 7 \times 7^3 = 7 \times 1 = 7 \)

From this, we can see that the remainders repeat every 3 terms:
- \( 7^1 \equiv 7 \)
- \( 7^2 \equiv 4 \)
- \( 7^3 \equiv 1 \)
- \( 7^4 \equiv 7 \) (and so on)

### Step 4: Determine the exponent modulo 3
Since the pattern repeats every 3 terms, we need to find \( 4444 \mod 3 \):
\[ 4444 \div 3 = 1481 \quad \text{(remainder 1)} \]
Thus, \( 4444 \equiv 1 \mod 3 \).

### Step 5: Find the corresponding power of 7
Since \( 4444 \equiv 1 \mod 3 \), we use the first term in our pattern:
\[ 7^{4444} \equiv 7^1 \equiv 7 \mod 9 \]

### Conclusion
The remainder when \( 4444^{4444} \) is divided by 9 is \( \boxed{7} \).

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