Ok, we want to see if the number

is divisible by 3.

Since

is in the units column, its value is

.

Since

is in the 10s column, its value is

.

Since

is in the 100s column, its value is

.

Continuing in this fashion we find that

is in the

s column, so its value is

.

Also note that every power of 10 will have one remainder when divided by 3 (since they can be written as 9 + 1, 99 + 1, 999 + 1, etc.

So the value of

is

Each of the

terms is divisible by 3, so the number will only be divisible by 3 if the sum of the remaining terms is divisible by 3.

So the number

is divisible by 3 if

is divisible by 3.