The remiander when `2^39` is divided by 39 is:

Options

A. 0
B. 2
C. 8
D. 1

Answer

To find the remainder when \( 2^{39} \) is divided by 39, we can use modular arithmetic and properties of exponents.

### Step-by-Step Solution:

1. **Express \( 2^{39} \) in a manageable form**:
We can express \( 2^{39} \) as \( (2^9)^4 \times 2^3 \). This is based on the property of exponents that states \( a^{m+n} = a^m \times a^n \).

\[ 2^{39} = (2^9)^4 \times 2^3 \]

**Hint**: Break down the exponent into smaller parts that are easier to calculate.

2. **Calculate \( 2^9 \)**:
First, we calculate \( 2^9 \):
\[ 2^9 = 512 \]

**Hint**: Use a calculator or compute \( 2^9 \) step by step (e.g., \( 2^3 = 8 \), \( 2^6 = 64 \), \( 2^9 = 64 \times 8 = 512 \)).

3. **Find \( 512 \mod 39 \)**:
Now, we need to find the remainder when \( 512 \) is divided by \( 39 \):
\[ 512 \div 39 \approx 13.128 \quad \text{(take the integer part, which is 13)} \]
\[ 39 \times 13 = 507 \]
\[ 512 - 507 = 5 \]
So, \( 512 \mod 39 = 5 \).

**Hint**: Use the formula \( a - b \times \text{(integer part of } (a/b)) \) to find the remainder.

4. **Substitute back into the expression**:
Now substitute back into our expression:
\[ 2^{39} \equiv (5^4) \times (2^3) \mod 39 \]

**Hint**: Remember that \( a \equiv b \mod m \) means \( a \) and \( b \) leave the same remainder when divided by \( m \).

5. **Calculate \( 5^4 \)**:
Now calculate \( 5^4 \):
\[ 5^4 = 625 \]

**Hint**: You can calculate \( 5^4 \) as \( (5^2)^2 = 25^2 = 625 \).

6. **Find \( 625 \mod 39 \)**:
Now find the remainder when \( 625 \) is divided by \( 39 \):
\[ 625 \div 39 \approx 16.025 \quad \text{(take the integer part, which is 16)} \]
\[ 39 \times 16 = 624 \]
\[ 625 - 624 = 1 \]
So, \( 625 \mod 39 = 1 \).

**Hint**: Again, use the same method as before to find the remainder.

7. **Combine results**:
Now we need to multiply this result by \( 2^3 \):
\[ 2^3 = 8 \]
So we have:
\[ (5^4 \mod 39) \times (2^3 \mod 39) \equiv 1 \times 8 \mod 39 \]
\[ \equiv 8 \mod 39 \]

**Hint**: When multiplying results under a modulus, you can simply multiply and then take the modulus of the product.

### Final Answer:
The remainder when \( 2^{39} \) is divided by 39 is \( \boxed{8} \).