<s>[INST] Factor $10^n -1$ There is an identity or quick way to factor problems of the form $c^x -1$ e.g. $10^n-1$, what is it? 
I mean that it can be represented in some way as multiple factors,e.g. $(x-a)(y-b)$, there is a way to expand the difference easily. 
 [/INST] Since it’s subtraction, you could use the difference of two squares I suppose.
$$a^2-b^2 = (a+b)(a-b) \implies a^c-b^c = \big(a^{\frac{c}{2}}+b^{\frac{c}{2}}\big)\big(a^{\frac{c}{2}}-b^{\frac{c}{2}}\big)$$
In this case, you have $b = 1$, so the factorization is
$$\big(c^{\frac{x}{2}}+1\big)\big(c^{\frac{x}{2}}-1\big) \implies (10^5+1)(10^5-1)$$
A more commonly used idea is factoring the difference of $n^{th}$ powers for all $n$ and the sum of $n^{th}$ powers for all odd $n$ (which isn’t relevant but is very similar to the former).
$$a^n-b^n = (a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^2+…+a^2b^{n-3}+ab^{n-2}+b^{n-1})$$
This works because through expansion because multiplying $a$ to each term is canceled by multiplying $b$ to the term before it, which leaves only the first term for $a$ and the last term for $b$.
$$ = \underbrace{a^n\color{blue}{+a^{n-1}b+a^{n-2}b^2+…+a^3b^{n-3}+a^2b^{n-2}+ab^{n-1}}}_{a(a^{n-1}+a^{n-2}b+…+ab^{n-2}+b^{n-1})}\underbrace{\color{red}{-a^{n-1}b-a^{n-2}b^2-a^{n-3}b^3-…-a^2b^{n-2}-ab^{n-1}}-b^n}_{-b(a^{n-1}+a^{n-2}b+…+ab^{n-2}+b^{n-1})}$$
$$= a^n-b^n$$
So, for instance, you can factor your example as follows:
$$10^n-1 = (10-1)(10^{n-1}+10^{n-2}+10^{n-3}+…+1)$$
which also bears striking resemblance to the geometric series if written as $\frac{10^n-1}{10-1} = 1+10+10^2+…+10^{n-1}$.
As a side note, you can use
$$(a+b)(a^{n-1}-a^{n-2}b+a^{n-3}b^2-a^{n-4}b^3+…+b^{n-1})$$
to factor the sum of $n^{th}$ powers, but this only works for an odd $n$.
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