The remainder obtained when `27^(50)` is divided by 12 is

Answer

To find the remainder when \( 27^{50} \) is divided by 12, we can follow these steps:

### Step 1: Simplify the base
First, we notice that \( 27 \) can be expressed in terms of \( 3 \):
\[ 27 = 3^3 \]
Thus, we can rewrite \( 27^{50} \) as:
\[ 27^{50} = (3^3)^{50} = 3^{150} \]

### Step 2: Find \( 3^{150} \mod 12 \)
Now, we need to find the remainder of \( 3^{150} \) when divided by 12. To do this, we can look for a pattern in the powers of \( 3 \) modulo \( 12 \).

Calculating the first few powers of \( 3 \):
- \( 3^1 = 3 \)
- \( 3^2 = 9 \)
- \( 3^3 = 27 \equiv 3 \mod 12 \)
- \( 3^4 = 81 \equiv 9 \mod 12 \)

We can see that:
- \( 3^1 \equiv 3 \mod 12 \)
- \( 3^2 \equiv 9 \mod 12 \)
- \( 3^3 \equiv 3 \mod 12 \)
- \( 3^4 \equiv 9 \mod 12 \)

### Step 3: Identify the pattern
From the calculations, we can observe a repeating pattern:
- When the exponent is odd, \( 3^{\text{odd}} \equiv 3 \mod 12 \)
- When the exponent is even, \( 3^{\text{even}} \equiv 9 \mod 12 \)

### Step 4: Determine the parity of the exponent
Since \( 150 \) is even, we can conclude:
\[ 3^{150} \equiv 9 \mod 12 \]

### Conclusion
Thus, the remainder when \( 27^{50} \) is divided by 12 is:
\[ \boxed{9} \]

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