Find the remainder when `3^2022` is divided by 5:

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the remainder when \( 3^{2022} \) is divided by 5, we can use Fermat's Little Theorem, which states that if \( p \) is a prime number and \( a \) is an integer not divisible by \( p \), then:

\[ a^{p-1} \equiv 1 \ (\text{mod} \ p) \]

In this case, let \( a = 3 \) and \( p = 5 \). Since 3 is not divisible by 5, we can apply the theorem.

### Step 1: Apply Fermat's Little Theorem
According to Fermat's Little Theorem:

\[ 3^{5-1} \equiv 1 \ (\text{mod} \ 5) \]

This simplifies to:

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

### Step 2: Reduce the exponent modulo 4
Next, we need to reduce the exponent 2022 modulo 4, since \( 3^4 \equiv 1 \).

\[ 2022 \mod 4 \]

Calculating \( 2022 \div 4 \):

\[ 2022 = 505 \times 4 + 2 \]

So, \( 2022 \mod 4 = 2 \).

### Step 3: Substitute back into the expression
Now we can rewrite \( 3^{2022} \) using the result from the previous step:

\[ 3^{2022} \equiv 3^2 \ (\text{mod} \ 5) \]

### Step 4: Calculate \( 3^2 \)
Now we calculate \( 3^2 \):

\[ 3^2 = 9 \]

### Step 5: Find the remainder when dividing by 5
Finally, we find the remainder of 9 when divided by 5:

\[ 9 \mod 5 = 4 \]

Thus, the remainder when \( 3^{2022} \) is divided by 5 is:

\[ \boxed{4} \]