If `lambda` is the remainder when `2^("2021")` is divided by 17, then the value of `lambda` must be equal to

Options

A. 3
B. 7
C. 13
D. 15

Answer

To find the remainder \( \lambda \) when \( 2^{2021} \) is divided by 17, 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:

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

In this case, \( a = 2 \) and \( p = 17 \).

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

\[ 2^{16} \equiv 1 \mod 17 \]

### Step 2: Reduce the exponent modulo \( p-1 \)
Now, we need to reduce \( 2021 \) modulo \( 16 \) (since \( 16 = 17 - 1 \)):

\[ 2021 \mod 16 \]

To find \( 2021 \mod 16 \), we can perform the division:

\[ 2021 \div 16 = 126.3125 \quad \text{(take the integer part, which is 126)} \]
\[ 126 \times 16 = 2016 \]
\[ 2021 - 2016 = 5 \]

So,

\[ 2021 \mod 16 = 5 \]

### Step 3: Substitute back into the exponent
Now we can replace \( 2021 \) in our original expression:

\[ 2^{2021} \equiv 2^5 \mod 17 \]

### Step 4: Calculate \( 2^5 \)
Now we calculate \( 2^5 \):

\[ 2^5 = 32 \]

### Step 5: Find \( 32 \mod 17 \)
Now we find the remainder when \( 32 \) is divided by \( 17 \):

\[ 32 \div 17 = 1.8824 \quad \text{(take the integer part, which is 1)} \]
\[ 1 \times 17 = 17 \]
\[ 32 - 17 = 15 \]

Thus,

\[ 32 \mod 17 = 15 \]

### Conclusion
Therefore, the value of \( \lambda \) is:

\[ \lambda = 15 \]

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