Simplify x−4=4(finite 7)

The term "finite 7" in this context typically refers to operations within the ring of integers modulo 7, denoted as $\mathbb{Z}_7$. This means all calculations are performed, and results are expressed, as remainders when divided by 7. The equation $x-4=4(\text{finite } 7)$ is interpreted as a congruence:

$$x - 4 \equiv 4 \pmod{7}$$

To isolate $x$, add 4 to both sides of the congruence:

$$x - 4 + 4 \equiv 4 + 4 \pmod{7}$$ $$x \equiv 8 \pmod{7}$$

To simplify $8 \pmod{7}$, we find the remainder when 8 is divided by 7.

$$8 = 1 \times 7 + 1$$

The remainder is 1. Therefore, $8 \equiv 1 \pmod{7}$.

Substituting this back into the congruence for $x$:

$$x \equiv 1 \pmod{7}$$

This means $x$ is any integer that leaves a remainder of 1 when divided by 7. For example, $x$ could be $1, 8, 15, -6$, and so on. If a single value from the set $\{0, 1, 2, 3, 4, 5, 6\}$ is expected, then $x=1$.