<s>[INST] What's the easiest way to factor $5^{10} - 1$? What's the easiest way to factor $5^{10} - 1$?
I believe $5 - 1$ is a factor based off the binomial theorem. 
From there I do not know. 
We are using congruence's in this class.
 [/INST] Note that
$$
5^{10} - 1 = (5^5+1)(5^5-1)
$$
And that
$$
5^5-1 = (5 - 1)(5^4 + 5^3 + 5^2 + 5 + 1)\\
5^5+1 = (5+1)(5^4 - 5^3 + 5^2 - 5 + 1)
$$
That should give you a fairly good start. We cannot factor this further using formal factorization alone.
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