Question

What is the remainder when (1234567890123456789)²⁴ is divided by 6561?

Answer 1

Notice that $6561=9^4$ and the number $1234567890123456789$ is divisible by $9$ as the sum of its digits =$2(1+2+\cdots+9)=2\left(\frac{9(9+1)}{2}\right)=90$ is divisible by $9$.

That means the number in question is divisible by $9^{24}$ and consequently the number is divisible by $9^4$ which is $6561$.