<s>[INST] Let $k = 2008^2 + 2^{2008}$. What is the last digit of $k^2 + 2^k$? Let $k = 2008^2 + 2^{2008}$. What is the last digit of $k^2 + 2^k.$
I thought of this
$$2008^2+2^{2008}\pmod{10} ≡ {-2}^2+{2^4}^{502}\pmod{10} ≡ 4+{-4}^{502}\pmod{10} ≡ 4+6^{251} \pmod{10}$$
but I still cannot prove it. Maybe there is a clever solution but so far I have been unable to spot it. Can anyone help me?
 [/INST] Since the cyclicity of $2$ is $2, 4, 8, 6$. It implies that for all $ n \in \mathbb{N}$.
$$2^{4n+1} \equiv 2 \mod 10 \\ 2^{4n+2} \equiv 4 \mod 10 \\ 2 ^{4n+3} \equiv 8 \mod 10 \\ 2^{4n+4} \equiv 6 \mod 10$$
Since $2008$ is of the form $4n$. So, $2^{2008} \equiv 6 \mod 10$.
But $2008 \equiv 8 \mod 10 \implies 2008^2 \equiv 8^2 \equiv 64 \equiv 4 \mod 10$.
So, $k = 2008^2 + 2^{2008} \equiv 4 + 6 \equiv 0 \mod 10 \implies k^2 \equiv 0 \mod 10$.
Now, 
$2^{2008} \equiv 0 \mod 4$ and $2008 \equiv 0 \mod 4 \implies 2008^2 \equiv 0 \mod 4$.
$$k = 2^{2008} + 2008^2 \equiv 0 + 0\equiv 0 \mod 4 $$
So, $k$ is of the form $4n$ for some positive integer $n$. So,
$$ 2^{k} \equiv 2^{4n} \equiv 6 \mod 10 $$
Therefore, $$2^{k} + k^{2} \equiv 6 + 0 \equiv 6 \mod 10$$
So, the last digit of $2^k + k^2$ is $6$.
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