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.