`(2021)^3762` is divided by 17 then find the remainder

Answer

To solve the problem of finding the remainder when \( (2021)^{3762} \) is divided by 17, we can follow these steps:

### Step 1: Find the remainder of 2021 when divided by 17
First, we calculate \( 2021 \mod 17 \).

\[ 2021 \div 17 \approx 118.882 \quad \text{(we take the integer part, which is 118)} \]
\[ 17 \times 118 = 2006 \]
\[ 2021 - 2006 = 15 \]

Thus, the remainder when 2021 is divided by 17 is 15.

### Step 2: Rewrite the expression using the remainder
Now we can rewrite the original expression \( (2021)^{3762} \) in terms of its remainder:

\[ (2021)^{3762} \equiv (15)^{3762} \mod 17 \]

### Step 3: Apply Fermat's Little Theorem
Fermat's Little Theorem states that if \( p \) is a prime and \( a \) is an integer not divisible by \( p \), then:

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

Here, \( p = 17 \) and \( a = 15 \). Since 15 is not divisible by 17, we can apply the theorem:

\[ 15^{16} \equiv 1 \mod 17 \]

### Step 4: Reduce the exponent modulo 16
Next, we need to reduce the exponent 3762 modulo 16:

\[ 3762 \div 16 = 235.125 \quad \text{(integer part is 235)} \]
\[ 16 \times 235 = 3760 \]
\[ 3762 - 3760 = 2 \]

Thus, \( 3762 \mod 16 = 2 \).

### Step 5: Calculate \( 15^{3762} \mod 17 \)
Now we can simplify our expression:

\[ 15^{3762} \equiv 15^2 \mod 17 \]

Calculating \( 15^2 \):

\[ 15^2 = 225 \]

Now we find \( 225 \mod 17 \):

\[ 225 \div 17 \approx 13.235 \quad \text{(integer part is 13)} \]
\[ 17 \times 13 = 221 \]
\[ 225 - 221 = 4 \]

### Final Answer
Thus, the remainder when \( (2021)^{3762} \) is divided by 17 is:

\[ \boxed{4} \]