Finding the Remainder of $7^{29} + 4$ Divided by 6

To find the remainder when $7^{29} + 4$ is divided by 6, we use modular arithmetic. We need to calculate $(7^{29} + 4) \pmod{6}$.

Step 1: Analyze the Base Modulo 6

First, find the remainder when the base, 7, is divided by 6.

$7 \div 6 = 1 \text{ with a remainder of } 1$

In modular arithmetic notation:

$7 \equiv 1 \pmod{6}$

Step 2: Apply the Exponent

Now, raise both sides of the congruence to the power of 29.

$7^{29} \equiv 1^{29} \pmod{6}$

Since $1$ raised to any power is $1$:

$7^{29} \equiv 1 \pmod{6}$

This means $7^{29}$ leaves a remainder of 1 when divided by 6.

Step 3: Add the Constant Term

Next, we incorporate the '+ 4' term from the original expression.

$7^{29} + 4 \equiv 1 + 4 \pmod{6}$

$7^{29} + 4 \equiv 5 \pmod{6}$

Step 4: Determine the Final Remainder

The expression $7^{29} + 4$ is congruent to 5 modulo 6.

Therefore, the remainder when $7^{29} + 4$ is divided by 6 is 5.