The remainder of `(888^(222) + 222^(888))/(5)` is :

Options

A. 0
B. 1
C. 3
D. 4

Answer

To find the remainder of \( \frac{888^{222} + 222^{888}}{5} \), we can use modular arithmetic. We will calculate \( 888^{222} \mod 5 \) and \( 222^{888} \mod 5 \) separately and then add the results together.

### Step 1: Calculate \( 888 \mod 5 \)

First, we find the remainder when 888 is divided by 5:

\[ 888 \div 5 = 177.6 \quad \text{(the integer part is 177)} \]
\[ 888 - (177 \times 5) = 888 - 885 = 3 \]

So, \( 888 \equiv 3 \mod 5 \).

### Step 2: Calculate \( 888^{222} \mod 5 \)

Now we need to calculate \( 3^{222} \mod 5 \). To do this, we can use Fermat's Little Theorem, which states that if \( p \) is a prime and \( a \) is not divisible by \( p \), then \( a^{p-1} \equiv 1 \mod p \).

Here, \( p = 5 \) and \( a = 3 \):

\[ 3^{4} \equiv 1 \mod 5 \]

Now we find \( 222 \mod 4 \):

\[ 222 \div 4 = 55.5 \quad \text{(the integer part is 55)} \]
\[ 222 - (55 \times 4) = 222 - 220 = 2 \]

Thus, \( 222 \equiv 2 \mod 4 \). Therefore:

\[ 3^{222} \equiv 3^{2} \mod 5 \]
\[ 3^{2} = 9 \equiv 4 \mod 5 \]

### Step 3: Calculate \( 222 \mod 5 \)

Next, we find the remainder when 222 is divided by 5:

\[ 222 \div 5 = 44.4 \quad \text{(the integer part is 44)} \]
\[ 222 - (44 \times 5) = 222 - 220 = 2 \]

So, \( 222 \equiv 2 \mod 5 \).

### Step 4: Calculate \( 222^{888} \mod 5 \)

Now we need to calculate \( 2^{888} \mod 5 \). Again, we can use Fermat's Little Theorem:

\[ 2^{4} \equiv 1 \mod 5 \]

Now we find \( 888 \mod 4 \):

\[ 888 \div 4 = 222 \quad \text{(exactly, so the remainder is 0)} \]

Thus, \( 888 \equiv 0 \mod 4 \). Therefore:

\[ 2^{888} \equiv 2^{0} \equiv 1 \mod 5 \]

### Step 5: Combine the results

Now we add the results from Steps 2 and 4:

\[ 888^{222} + 222^{888} \equiv 4 + 1 \mod 5 \]
\[ 4 + 1 = 5 \equiv 0 \mod 5 \]

### Final Answer

The remainder of \( \frac{888^{222} + 222^{888}}{5} \) is \( \boxed{0} \).

---