Show that that the number 3^(2,012)+3^(2,009) is divisble by 12

To show that \( 3^{2012} + 3^{2009} \) is divisible by 12, we will break the proof into steps, examining divisibility by both 3 and 4 (since \( 12 = 3 \times 4 \)).

### Step 1: Check divisibility by 3
We start by checking the expression \( 3^{2012} + 3^{2009} \) modulo 3. Since powers of 3 are clearly divisible by 3, both \( 3^{2012} \) and \( 3^{2009} \) are divisible by 3. Therefore:
\[
3^{2012} + 3^{2009} \equiv 0 + 0 \equiv 0 \pmod{3}
\]
Thus, \( 3^{2012} + 3^{2009} \) is divisible by 3.

### Step 2: Check divisibility by 4
Now, we need to check if \( 3^{2012} + 3^{2009} \) is divisible by 4. To do this, we first observe the behavior of powers of 3 modulo 4:
\[
3^1 \equiv 3 \pmod{4}, \quad 3^2 \equiv 1 \pmod{4}, \quad 3^3 \equiv 3 \pmod{4}, \quad 3^4 \equiv 1 \pmod{4}
\]
So, the powers of 3 modulo 4 alternate between 3 and 1. Specifically:
- If the exponent is odd, \( 3^n \equiv 3 \pmod{4} \).
- If the exponent is even, \( 3^n \equiv 1 \pmod{4} \).

Now, check the two terms in the expression:
- \( 3^{2012} \): Since 2012 is even, \( 3^{2012} \equiv 1 \pmod{4} \).
- \( 3^{2009} \): Since 2009 is odd, \( 3^{2009} \equiv 3 \pmod{4} \).

Thus:
\[
3^{2012} + 3^{2009} \equiv 1 + 3 \equiv 4 \equiv 0 \pmod{4}
\]
Therefore, \( 3^{2012} + 3^{2009} \) is divisible by 4.

### Step 3: Conclusion
Since \( 3^{2012} + 3^{2009} \) is divisible by both 3 and 4, it is divisible by \( \text{lcm}(3, 4) = 12 \).

Thus, we have shown that \( 3^{2012} + 3^{2009} \) is divisible by 12.

Would you like further details on any part of this explanation? 

Here are some related questions you might find interesting:
1. How do we determine the pattern of powers of 3 modulo different numbers?
2. Can this method be used to prove divisibility by other numbers?
3. What are some examples of numbers divisible by both 3 and 4?
4. How would the process differ for other bases, such as 2 or 5?
5. How can modular arithmetic help solve other divisibility problems?

**Tip:** Modular arithmetic is a powerful tool for checking divisibility by specific numbers. It's helpful to identify periodic patterns in powers of numbers modulo smaller divisors like 3, 4, and 5.

Show that the number 3^(2,012) + 3^(2,009) is divisible by 12

Learn how to prove that 3^(2012) + 3^(2009) is divisible by 12 using modular arithmetic and number theory. This problem covers key concepts like divisibility rules, powers of numbers modulo 4, and LCM properties. Suitable for students in Grades 9-12.

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