<s>[INST] Smallest Prime-Factor of $4^{52} + 52^{2013} + 2013^{52}$ I would like to find out the smallest prime factor of $4^{52} + 52^{2013} + 2013^{52}$ by hand.
Thanks in advance
 [/INST] The given number is odd, so it cannot be divided by $2$.
In mod $3$, we have
$$\begin{align}4^{52}+52^{2013}+2013^{52}\equiv 1+1+0\equiv 2\end{align}$$
In mod $5$, we have
$$\begin{align}4^{52}+52^{2013}+2013^{52}&\equiv (-1)^{52}+2^{2013}+3^{52}\\&\equiv 1+2\cdot 2^{2012}+(-1)^{26}\\&\equiv 1+2\cdot {16}^{503}+1\\&\equiv 1+2+1\\&\equiv 4\end{align}$$
In mod $7$, we have
$$\begin{align}4^{52}+52^{2013}+2013^{52}&\equiv (-3)^{52}+3^{2013}+4^{52}\\&\equiv \color{red}{2^{26}}+3\cdot 2^{1006}+\color{red}{2^{26}}\\&\equiv \color{red}{2^{27}}+3\cdot 4^{503}\\&\equiv 2\cdot 4^{13}+3\cdot 4\cdot {2}^{251}\\&\equiv  2\cdot 4\cdot {2}^6+5\cdot 2^{251}\\& \equiv 1\cdot 2^6+5\cdot 2\cdot 4^{125}\\&\equiv 64+3\cdot 4\cdot 2^{62}\\&\equiv 1+5\cdot 4^{31}\\&\equiv 1+5\cdot 4\cdot 2^{15}\\&\equiv 1+6\cdot 2\cdot {4}^{7}\\&\equiv 1+5\cdot 4\cdot 2^{3}\\&\equiv 1+6\cdot 1\\&\equiv 0 \end{align}$$ 
Hence, the answer is $7$.
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