<s>[INST] Sum of the digits of $N=5^{2012}$ The sum of the digits of $N=5^{2012}$ is computed.
The sum of the digits of the resulting sum is then computed.
The process of computing the sum is repeated until a single digit number is obtained.
What is this single digit number?
 [/INST] You want to know the value of $5^{2012} \pmod 9$.
Since $\varphi(9) = 3^2 - 3 = 6$ and $\gcd(5,9) = 1$, then, by Euler's theorem,  $5^6 \equiv 1 \pmod 9$.
Since $2012 = 335 \times 6 + 2$, 
$$5^{2012} \equiv (5^6)^{335} \times 5^2 
\equiv 1^{335} \times 25 \equiv 7 \pmod 9.$$
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