If $A(z_1), B(z_2), C(z_3)$ are the vertices of triangle such that $z_1 = \frac{z_2 - iz_1}{1 - i}$, $|z_1| = 3$, $|z_2| = 4$ and $|z_2 + iz_1| = |z_1| + |z_2|$, then area of $\triangle ABC$ is ____.
Step-by-Step Solution
Key Concept: Unit digit cycles mod 4; target pairs sum to 10, and counting uses the Chinese Remainder Theorem structure.
The unit digit of $7^k$ cycles as $7, 9, 3, 1$ with period 4. For $7^p + 7^q$ to end in 0, we need the unit digits to sum to 10. This occurs when $p \equiv 1 \pmod{4}$ and $q \equiv 2 \pmod{4}$ (giving digits 7 and 3), and three other similar pairings. For 100 choices of $p$ from 0 to 99, there are 25 choices each of $p \equiv 1, 2, 3, 0 \pmod{4}$. With 4 valid $(p \bmod 4, q \bmod 4)$ pairs and 25 choices per pair, there are $100 \times 25 = 2500$ ordered pairs.
Correct Answer: 6.25