What is the remainder when `(5555)^(2222)+(2222)^(5555)` is divided by 7 ?

Answer

To find the remainder when \( (5555)^{2222} + (2222)^{5555} \) is divided by 7, we can follow these steps:

### Step 1: Find the remainder of 5555 when divided by 7.
To do this, we can perform the division:
\[ 5555 \div 7 = 793 \quad \text{(quotient)} \]
Calculating \( 7 \times 793 \):
\[ 7 \times 793 = 5551 \]
Now, subtract this from 5555 to find the remainder:
\[ 5555 - 5551 = 4 \]
Thus, \( 5555 \equiv 4 \mod 7 \).

### Step 2: Find the remainder of 2222 when divided by 7.
Similarly, we divide:
\[ 2222 \div 7 = 317 \quad \text{(quotient)} \]
Calculating \( 7 \times 317 \):
\[ 7 \times 317 = 2219 \]
Now, subtract this from 2222 to find the remainder:
\[ 2222 - 2219 = 3 \]
Thus, \( 2222 \equiv 3 \mod 7 \).

### Step 3: Rewrite the expression using the remainders.
Now we can rewrite the original expression using the remainders we found:
\[ (5555)^{2222} + (2222)^{5555} \equiv (4)^{2222} + (3)^{5555} \mod 7 \]

### Step 4: Calculate \( 4^{2222} \mod 7 \).
Using Fermat's Little Theorem, since 7 is prime:
\[ a^{p-1} \equiv 1 \mod p \]
For \( a = 4 \) and \( p = 7 \):
\[ 4^{6} \equiv 1 \mod 7 \]
Now, we need to find \( 2222 \mod 6 \) (since \( 6 = 7-1 \)):
\[ 2222 \div 6 = 370 \quad \text{(quotient)} \]
Calculating \( 6 \times 370 = 2220 \):
\[ 2222 - 2220 = 2 \]
Thus, \( 2222 \equiv 2 \mod 6 \). Therefore:
\[ 4^{2222} \equiv 4^{2} \mod 7 \]
Calculating \( 4^{2} \):
\[ 4^{2} = 16 \quad \text{and} \quad 16 \mod 7 = 2 \]
So, \( 4^{2222} \equiv 2 \mod 7 \).

### Step 5: Calculate \( 3^{5555} \mod 7 \).
Again using Fermat's Little Theorem:
\[ 3^{6} \equiv 1 \mod 7 \]
Now, we need to find \( 5555 \mod 6 \):
\[ 5555 \div 6 = 925 \quad \text{(quotient)} \]
Calculating \( 6 \times 925 = 5550 \):
\[ 5555 - 5550 = 5 \]
Thus, \( 5555 \equiv 5 \mod 6 \). Therefore:
\[ 3^{5555} \equiv 3^{5} \mod 7 \]
Calculating \( 3^{5} \):
\[ 3^{5} = 243 \quad \text{and} \quad 243 \mod 7 = 5 \]
So, \( 3^{5555} \equiv 5 \mod 7 \).

### Step 6: Combine the results.
Now we can combine the results:
\[ (5555)^{2222} + (2222)^{5555} \equiv 2 + 5 \mod 7 \]
Calculating:
\[ 2 + 5 = 7 \quad \text{and} \quad 7 \mod 7 = 0 \]

### Final Answer:
The remainder when \( (5555)^{2222} + (2222)^{5555} \) is divided by 7 is \( \boxed{0} \).