Find the remainder when `(3)^(152)` is divided by 15.

Options

A. 3
B. 2
C. 6
D. 9

Answer

To find the remainder when \( 3^{152} \) is divided by 15, we can use the concept of modular arithmetic and the properties of powers.

### Step 1: Identify the pattern of \( 3^n \mod 15 \)

First, let's calculate the first few powers of 3 modulo 15:

- \( 3^1 = 3 \)
- \( 3^2 = 9 \)
- \( 3^3 = 27 \equiv 12 \mod 15 \)
- \( 3^4 = 81 \equiv 6 \mod 15 \)
- \( 3^5 = 243 \equiv 3 \mod 15 \)

### Step 2: Observe the cycle

From the calculations, we can see that the powers of 3 modulo 15 repeat every 4 terms:

- \( 3^1 \equiv 3 \)
- \( 3^2 \equiv 9 \)
- \( 3^3 \equiv 12 \)
- \( 3^4 \equiv 6 \)
- \( 3^5 \equiv 3 \) (and the cycle repeats)

### Step 3: Determine the equivalent exponent

Since the pattern repeats every 4 terms, we can find \( 3^{152} \mod 15 \) by finding the equivalent exponent of 152 modulo 4:

\[ 152 \mod 4 = 0 \]

This means \( 3^{152} \) corresponds to \( 3^0 \) in the cycle.

### Step 4: Find the value of \( 3^0 \mod 15 \)

From the cycle, we know:

- \( 3^0 \equiv 1 \mod 15 \)

### Step 5: Conclusion

Thus, the remainder when \( 3^{152} \) is divided by 15 is:

\[ \text{Remainder} = 1 \]