how would I go about calculating 147^136 mod(12)

To calculate \( 147^{136} \mod 12 \), you can simplify the problem using modular arithmetic properties to avoid calculating large numbers. Here's a step-by-step method:

### Step 1: Simplify \( 147 \mod 12 \)
We first reduce \( 147 \mod 12 \):
\[
147 \div 12 = 12 \times 12 + 3
\]
So:
\[
147 \equiv 3 \pmod{12}
\]
This means \( 147 \mod 12 \equiv 3 \mod 12 \).

### Step 2: Reduce the exponentiation to \( 3^{136} \mod 12 \)
Now, the problem becomes:
\[
147^{136} \mod 12 \equiv 3^{136} \mod 12
\]

### Step 3: Use exponentiation properties with mod
We'll calculate \( 3^{136} \mod 12 \) by finding a repeating pattern in powers of 3 modulo 12.

#### Calculate powers of 3 mod 12:
\[
3^1 \mod 12 = 3
\]
\[
3^2 \mod 12 = 9
\]
\[
3^3 \mod 12 = 27 \mod 12 = 3
\]
\[
3^4 \mod 12 = 81 \mod 12 = 9
\]
The powers of 3 mod 12 repeat every two terms: \( 3, 9, 3, 9, \ldots \).

### Step 4: Use the pattern to simplify \( 3^{136} \mod 12 \)
The pattern repeats every two terms, so we divide the exponent 136 by 2:
\[
136 \div 2 = 68
\]
Since 136 is even, \( 3^{136} \mod 12 \) follows the pattern of \( 3^2 \mod 12 \), which is:
\[
3^{136} \mod 12 = 9
\]

### Final Answer:
\[
147^{136} \mod 12 = 9
\]

Would you like more details or have any questions?

Here are some related questions to deepen your understanding:
1. What is the general rule for simplifying powers in modular arithmetic?
2. How do we use Fermat's Little Theorem to simplify mod calculations for primes?
3. How would you calculate \( 147^{137} \mod 12 \)?
4. Why is finding patterns in powers of numbers useful in modular arithmetic?
5. What if the modulus was 7 instead of 12—how would the solution change?

**Tip:** In modular arithmetic, look for cycles or repeating patterns when calculating large powers modulo a number!

Learn how to simplify and calculate large exponentiation problems like 147^136 mod 12 using modular arithmetic. This step-by-step guide demonstrates techniques such as reducing the base, detecting patterns in powers, and applying modular exponentiation. Suitable for students in Grades 10-12.

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