Find the remainder

$\begin{aligned} 8^{2048}-6^{2048} & = (8^{1024}-6^{1024})(8^{1024}+6^{1024}) \text{ (mod 700)} \\ & = (8^{512}-6^{512})(8^{512}+6^{512})(8^{1024}+6^{1024}) \text{ (mod 700)} \\ & = (8^{256}-6^{256})(8^{256}+6^{256})(8^{512}+6^{512})(8^{1024}+6^{1024}) \text{ (mod 700)} \\ & = (8-6)(8+6)(8^2+6^2) (8^3+6^3)...(8^{1024}+6^{1024}) \text{ (mod 700)} \\ & = (2){\color{#3D99F6}(14)(100)} (8^3+6^3)...(8^{1024}+6^{1024}) \text{ (mod 700)} \\ & = {\color{#3D99F6}\boxed{0}} \text{ (mod 700)} \end{aligned}$

Yes , $8^{2048}-6^{2048}$ is divisible by 700.

Apply the formula a^2n - b^2n = (a^n - b^n) (a^n + b^n) repeatedly until the power of 8 and 6 is 1. 2048 = 2^20.

8^2048 - 6^2048 = (8^1024 - 6^1024) * (8^1024 +6^1024) = (8^516 - 6^516) * (8^516 + 6^516) * (8^1024 +6^1024) = (8^256 - 6^256) * (8^256 - 6^256) *(8^516 + 6^516) * (8^1024 +6^1024) ..... = (8^2 - 6^2) * (8^2 +6^2) * ... * (8^256 - 6^256) *(8^516 + 6^516) * (8^1024 +6^1024) = (8 - 6) * (8 + 6) * (8^2 +6^2) * ... * (8^256 - 6^256) *(8^516 + 6^516) * (8^1024 +6^1024)

8+6 =14 (8^2 +6^2) = 64 + 36 = 100

2, 7, 100 are factors of 8^2048 - 6^2048 Therefore 700 divides 8^2048 - 6^2048

We have:

$8^{2048}-6^{2048} \equiv 0-2^{2048} \cdot 3^{2048} \equiv -4^{1024} \cdot 3^{2048} \equiv 0 \pmod 4$

$8^{2048}-6^{2048} \equiv 1^{2048}-(-1)^{2048} \equiv 1 - 1 \equiv 0 \pmod 7$

$\begin{aligned} 8^{2048}-6^{2048} &\equiv 2^{6144 \mod \phi(25)}-6^{2048 \mod \phi(25)} \equiv 2^{6144 \mod 20}-6^{2048 \mod 20} \pmod{25}\\ &\equiv 2^4-6^8 \equiv 16-11^4 \equiv 16-21^2 \equiv 16 - 16 \equiv 0 \pmod {25} \end{aligned}$

So, $8^{2048}-6^{2048} \equiv 0 \pmod{700}$ , and we conclude that $8^{2048}-6^{2048}$ is divisible by $700$ .

$8^{2048} - 6^{2048} \\ = 2^{3 \times 2048} - 2^{2048} \times 3^{2048} \\ = 2^{2048} [ 2^{2 \times 2048} - 3^{2048} ] \\ \text{Now we eliminate all }a^n + b^n \text{ as they won't simply be said to divide 7. We'll take and reduce all numbers of the form } a^n - b^n = 4^{2048} - 3^{2048} \\ = (4^{1024} + 3^{1024})(4^{1024} - 3^{1024}) ~~ \boxed{2048 = 2 \times 1024}\\ = 4^{512} - 3^{512} ~~ \boxed{1024 = 2 \times 512}\\ = 4^{256} - 3^{256} ~~ \boxed{512 = 2 \times 256}\\ =4^{128} - 3^{128} ~~ \boxed{256 = 2 \times 128}\\ = 4^{64} - 3^{64} ~~ \boxed{128 = 2 \times 64}\\ = 4^{32} - 3^{32} ~~ \boxed{64 = 2 \times 32}\\ = 4^{16} - 3^{16} ~~ \boxed{32 = 2 \times 16}\\ = 4^{8} - 3^{8} ~~ \boxed{16 = 2 \times 8}\\ = 4^{4} - 3^{4} ~~ \boxed{8 = 2 \times 4}\\ = 4^{2} - 3^{2} ~~ \boxed{4 = 2 \times 2}\\ \boxed{\implies (4+3)(4-3) \implies 7 * 1 = 7}$

Even simpler using congruency.

$8 \equiv 1 (\mathrm{mod} \ 7) \\ 8^{2048} \equiv 1^{2048} \equiv 1 (\mathrm{mod} \ 7) \\ 6 \equiv -1 (\mathrm{mod} \ 7) \\ 6^{2048} \equiv -1^{2048} \equiv 1 (\mathrm{mod} \ 7) \\ 8^{2048} - 6^{2048} \equiv 1 - 1 = 0 (\mathrm{mod} \ 7) \\ \boxed{\therefore 7 | (8^{2048} - 6^{2048})}$