Finding a Remainder when Dividing by 3, 9, etc...

To find the remainder of a number when divided by 3 or 9, you can use the sum of its digits. The remainder of the number is the same as the remainder of the sum of its digits when divided by 3 or 9.

For example, to find the remainder of $34952 \div 3$:
Sum the digits: $3+4+9+5+2 = 23$.
Find the remainder of the sum: $23 \div 3$ gives a remainder of 2. So, the remainder of $34952 \div 3$ is 2.

For very large numbers, you can use modular techniques by finding the remainder of each digit individually (or groups of digits) and summing them. For example, when dividing by 3, any 3, 6, or 9 can be treated as 0.