Funny exponent (Part 2)

$\begin{aligned} 3^{11,000,001} & \equiv 3 \times 3^{2 \times 5,500,000} \text{(mod }10^7) \\ & \equiv 3 \times 9^{5,500,000} \text{(mod }10^7) \\ & \equiv 3 (10-1)^{5,500,000} \text{(mod }10^7) \\ & \equiv 3 ( \cdots + 5,500,000(5,499,999)(50) - 55,000,000+ 1) \text{(mod }10^7) \\ & \equiv 3 ( \cdots + 55,000,000(27,499,995-1)+ 1) \text{(mod }10^7) \\ & \equiv 3 ( \cdots + 55,000,000(27,499,994)+ 1) \text{(mod }10^7) \\ & \equiv 3 ( \cdots + 110,000,000(13,749,997)+ 1) \text{(mod }10^7) \\ & \equiv \boxed 3 \text{(mod }10^7) \end{aligned}$

The Carmichael's lambda function of $10^7$ is $5\times 10^5$ .

Thus, $3^{5\times10^5} \equiv 1 \pmod{10^7}$ . Raise both sides to the power of 22 yields $3^{11\times10^6} \equiv 1 \pmod{10^7}.$ Now multiply both sides by 3, and the answer follows.