prove that 7 to the power n + 4 to the power n is divided by 3

Answer 1

To prove that \( 7^n + 4^n \) is divisible by 3 for any integer \( n \), we can check the expression modulo 3.

Step 1: Check \( 7 \mod 3 \)

\( 7 \) can be reduced modulo 3: \[ 7 \mod 3 = 1 \] Thus, \[ 7^n \mod 3 = 1^n \mod 3 = 1 \]

Step 2: Check \( 4 \mod 3 \)

Next, we consider \( 4 \mod 3 \): \[ 4 \mod 3 = 1 \] Thus, \[ 4^n \mod 3 = 1^n \mod 3 = 1 \]

Step 3: Combine the results

Now, we can combine the results: \[ 7^n + 4^n \equiv 1 + 1 \mod 3 \] This simplifies to: \[ 7^n + 4^n \equiv 2 \mod 3 \]

Conclusion

The expression \( 7^n + 4^n \) is congruent to 2 modulo 3, which means it is not divisible by 3 (it would need to be congruent to 0 modulo 3 to be divisible by 3).

Let's correct the statement. The proposition should be that \( 7^n + 4^n \equiv 2 \mod 3 \).

Thus, the claim that \( 7^n + 4^n \) is divisible by 3 is not true; rather, \( 7^n + 4^n \equiv 2 \mod 3\).