number theory - 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?

Answer

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 $2012 = 335 \times 6 + 2$,

$$5^{2012} \equiv (5^6)^{335} \times 5^2 \equiv 1^{335} \times 25 \equiv 7 \pmod 9.$$