Find the remainder when `123^(321)` is divided by 5.

Answer

To find the remainder when \( 123^{321} \) is divided by 5, we can follow these steps:

### Step 1: Simplify the base modulo 5
First, we need to simplify \( 123 \) modulo \( 5 \):
\[ 123 \div 5 = 24 \quad \text{(whole part)} \]
Calculating the remainder:
\[ 123 - (5 \times 24) = 123 - 120 = 3 \]
So, \( 123 \equiv 3 \mod 5 \).

### Step 2: Rewrite the expression
Now we can rewrite the original expression using the simplified base:
\[ 123^{321} \equiv 3^{321} \mod 5 \]

### Step 3: Use Euler's theorem
Next, we can apply Euler's theorem. First, we find \( \phi(5) \):
\[ \phi(5) = 5 - 1 = 4 \]
According to Euler's theorem, since \( 3 \) and \( 5 \) are coprime:
\[ 3^{\phi(5)} \equiv 1 \mod 5 \]
This means:
\[ 3^4 \equiv 1 \mod 5 \]

### Step 4: Reduce the exponent modulo \( \phi(5) \)
Now we need to reduce the exponent \( 321 \) modulo \( 4 \):
\[ 321 \div 4 = 80 \quad \text{(whole part)} \]
Calculating the remainder:
\[ 321 - (4 \times 80) = 321 - 320 = 1 \]
So, \( 321 \equiv 1 \mod 4 \).

### Step 5: Substitute back into the expression
Now we can substitute back into our expression:
\[ 3^{321} \equiv 3^{1} \mod 5 \]

### Step 6: Calculate the final result
Finally, we calculate:
\[ 3^{1} \equiv 3 \mod 5 \]

Thus, the remainder when \( 123^{321} \) is divided by \( 5 \) is \( \boxed{3} \).

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