$4^7 - 4$ Expression Factorization

The expression given is $4^7 - 4$. We first factorize it:

$4^7 - 4 = 4 \times (4^6 - 1)$

$4^7 - 4$ Divisibility by 2

Since $4^7 - 4 = 4(4^6 - 1)$, the expression is a multiple of 4. Any multiple of 4 is also a multiple of 2. Thus, $4^7 - 4$ is divisible by 2.

$4^7 - 4$ Divisibility by 4

From the factorization $4(4^6 - 1)$, it is evident that the expression contains a factor of 4. Thus, $4^7 - 4$ is divisible by 4.

$4^7 - 4$ Divisibility by 7

To check divisibility by 7, we evaluate the expression modulo 7:

$4^3 \equiv 1 \pmod{7}$

$4^7 = 4^{3 \times 2 + 1} = (4^3)^2 \times 4^1 \equiv 1^2 \times 4 \equiv 4 \pmod{7}$

Therefore, $4^7 - 4 \equiv 4 - 4 \equiv 0 \pmod{7}$.

This confirms that $4^7 - 4$ is divisible by 7.

$4^7 - 4$ Divisibility by 8

Consider the factored form $4(4^6 - 1)$.

$4^6 = (2^2)^6 = 2^{12}$.

The expression is $4(2^{12} - 1)$.

$2^{12} - 1$ is an odd number.

Let $2^{12} - 1 = k$, where $k$ is odd. The expression is $4k$.

A number of the form $4 \times \text{odd}$ is never divisible by 8. For $4k$ to be divisible by 8, $k$ would need to be even ($4k = 8m \implies k=2m$), which contradicts $k$ being odd.

Thus, $4^7 - 4$ is NOT a multiple of 8.

$4^7 - 4$ Final Conclusion on Multiples

The expression $4^7 - 4$ is confirmed to be a multiple of 2, 4, and 7. However, it is not a multiple of 8.

Therefore, 8 is the number that is NOT a multiple of $4^7 - 4$.