What is the remainder of `6^(36)/215`?

Options

A. 0
B. 1
C. 2
D. none of these

Answer

To find the remainder of \( 6^{36} \) when divided by 215, we can use modular arithmetic. Here’s a step-by-step solution:

### Step 1: Factor 215
First, we factor 215 into its prime factors:
\[ 215 = 5 \times 43 \]

### Step 2: Use the Chinese Remainder Theorem
To find \( 6^{36} \mod 215 \), we will find \( 6^{36} \mod 5 \) and \( 6^{36} \mod 43 \) separately, and then combine the results.

### Step 3: Calculate \( 6^{36} \mod 5 \)
Calculating \( 6 \mod 5 \):
\[ 6 \equiv 1 \mod 5 \]
Thus,
\[ 6^{36} \equiv 1^{36} \equiv 1 \mod 5 \]

### Step 4: Calculate \( 6^{36} \mod 43 \)
Next, we calculate \( 6^{36} \mod 43 \). We can use Fermat's Little Theorem, which states that if \( p \) is a prime and \( a \) is not divisible by \( p \), then:
\[ a^{p-1} \equiv 1 \mod p \]
Here, \( p = 43 \) and \( a = 6 \):
\[ 6^{42} \equiv 1 \mod 43 \]
Now, we need to reduce \( 36 \) modulo \( 42 \):
\[ 36 < 42 \quad \text{(so we can use 36 directly)} \]
Now we compute \( 6^{36} \mod 43 \). We can calculate powers of \( 6 \) modulo \( 43 \):
- \( 6^1 \equiv 6 \)
- \( 6^2 \equiv 36 \)
- \( 6^3 \equiv 6 \times 36 \equiv 216 \mod 43 \equiv 1 \)

Since \( 6^3 \equiv 1 \mod 43 \), we can express \( 6^{36} \) as:
\[ 6^{36} = (6^3)^{12} \equiv 1^{12} \equiv 1 \mod 43 \]

### Step 5: Combine Results Using the Chinese Remainder Theorem
Now we have:
\[ 6^{36} \equiv 1 \mod 5 \]
\[ 6^{36} \equiv 1 \mod 43 \]
Since both congruences give us the same result, we can conclude:
\[ 6^{36} \equiv 1 \mod 215 \]

### Final Answer
Thus, the remainder when \( 6^{36} \) is divided by 215 is:
\[ \boxed{1} \]