If $13^n - 10^8$ is divided by 162, then the remainder is
Step-by-Step Solution
Key Concept: The binomial theorem applied to expressions like $(1+a)^n - (1+b)^n$ reveals divisibility patterns through coefficient analysis.
We expand $13^{99} - 19^{99} = (1 + 12)^{99} - (1 + 18)^{99}$. Since both bases are odd, both expansions are odd minus odd. Using the binomial theorem: $13^{99} = 1 + 99 \times 12 + \binom{99}{2}12^2 + ...$, and $19^{99} = 1 + 99 \times 18 + \binom{99}{2}18^2 + ...$. Computing the difference and checking divisibility by 162: the expression simplifies to $81(27 + ...)$ which is divisible by 81. Further analysis shows $13^{99} - 19^{99} \equiv 0 \pmod{162}$, so the remainder is 0.
Correct Answer: 0