Find the remainder `( 16^(200) + 5)/( 15)`
(a)6
(b)7
(c)10
(d)12

Options

A. 6
B. 7
C. 10
D. 12

Answer

To solve the problem of finding the remainder when \( \frac{16^{200} + 5}{15} \), we can follow these steps:

### Step 1: Simplify \( 16^{200} \mod 15 \)

First, we need to find the value of \( 16^{200} \) modulo \( 15 \). Notice that:

\[ 16 \equiv 1 \mod 15 \]

This is because when you divide \( 16 \) by \( 15 \), the remainder is \( 1 \).

### Step 2: Calculate \( 16^{200} \mod 15 \)

Since \( 16 \equiv 1 \mod 15 \), we can raise both sides to the power of \( 200 \):

\[ 16^{200} \equiv 1^{200} \mod 15 \]

Calculating \( 1^{200} \):

\[ 1^{200} = 1 \]

Thus, we have:

\[ 16^{200} \equiv 1 \mod 15 \]

### Step 3: Add 5 to the result

Now we add \( 5 \) to \( 16^{200} \):

\[ 16^{200} + 5 \equiv 1 + 5 \mod 15 \]

Calculating this gives:

\[ 1 + 5 = 6 \]

### Step 4: Find the remainder when divided by 15

Now, we need to find the remainder when \( 6 \) is divided by \( 15 \):

\[ 6 \mod 15 = 6 \]

### Conclusion

Thus, the remainder when \( 16^{200} + 5 \) is divided by \( 15 \) is \( 6 \).

The correct answer is (a) 6.

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