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

Answer

To find the remainder when \( 4444^{4444} \) is divided by 9, we can follow these steps:

### Step 1: Find the remainder of \( 4444 \) when divided by \( 9 \)

To find \( 4444 \mod 9 \), 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 original expression using the remainder

Now we can rewrite \( 4444^{4444} \mod 9 \) as:
\[ 4444^{4444} \equiv 7^{4444} \mod 9 \]

### Step 3: Determine the pattern in the powers of \( 7 \mod 9 \)

Next, we will calculate the first few powers of \( 7 \) modulo \( 9 \):
- \( 7^1 \equiv 7 \mod 9 \)
- \( 7^2 \equiv 49 \mod 9 \equiv 4 \mod 9 \)
- \( 7^3 \equiv 7 \times 7^2 \equiv 7 \times 4 = 28 \mod 9 \equiv 1 \mod 9 \)

From this, we see a repeating cycle every 3 powers:
- \( 7^1 \equiv 7 \)
- \( 7^2 \equiv 4 \)
- \( 7^3 \equiv 1 \)

### Step 4: Determine the exponent modulo \( 3 \)

Since the powers of \( 7 \mod 9 \) repeat every 3, we need to find \( 4444 \mod 3 \):
\[ 4444 \div 3 = 1481 \quad \text{(remainder 1)} \]
Thus,
\[ 4444 \equiv 1 \mod 3 \]

### Step 5: Use the result to find \( 7^{4444} \mod 9 \)

Since \( 4444 \equiv 1 \mod 3 \), we can find:
\[ 7^{4444} \equiv 7^1 \equiv 7 \mod 9 \]

### Final Answer

Thus, the remainder when \( 4444^{4444} \) is divided by \( 9 \) is:
\[ \boxed{7} \]