I want to find the remainder when a large number is divided by 9, so I think of a divisibility rule for 9 that might help. I recall that a number is divisible by 9 if and only if the sum of its digits is divisible by 9. So, I can try to find the sum of the digits of $8\cdot10^{18}+1^{18}$ and see what the remainder is when that is divided by 9. To find the sum of the digits, I need to write the number in decimal form. I notice that $8\cdot10^{18}$ is just 8 followed by 18 zeros, and $1^{18}$ is just 1, so the decimal form of the number is 8,000,000,000,000,000,001. Now, the sum of the digits is just 8 + 1 = 9, which is divisible by 9. That means the original number is also divisible by 9, and the remainder is 0. 
Find the remainder when $8\cdot10^{18}+1^{18}$ is divided by 9.