The question asks for the remainder when the expression $7^6 + 5$ is divided by 6. We can solve this using modular arithmetic.

Remainder Calculation for $7^6 + 5$ Divided by 6

We need to find the value of $7^6 + 5 \pmod{6}$.

1. First, consider the base, 7, in modulo 6.

   Since $7 = 1 \times 6 + 1$, we have $7 \equiv 1 \pmod{6}$.

2. Next, apply this to the term $7^6$. Using the property of modular arithmetic that if $a \equiv b \pmod{m}$, then $a^n \equiv b^n \pmod{m}$, we get:

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

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

3. Now, substitute this result back into the original expression:

   $7^6 + 5 \pmod{6}$

   Substitute $7^6 \equiv 1 \pmod{6}$: $1 + 5 \pmod{6}$

4. Calculate the final sum:

   $1 + 5 = 6$

   Since $6 = 1 \times 6 + 0$, we have $6 \equiv 0 \pmod{6}$.

Therefore, the remainder when $7^6 + 5$ is divided by 6 is 0.