I'm working on a little exercise I found in my high school book (printed in 2007) which is pretty complicated.

Is the sum of digits of 3^1000 a multiple of 7?

Do you have any advice to solve this type of problem (without programming of course!)

Thanks a lot!!

PS :

We are a group of 3 french people working on it since 2007.

The sum calculated with Python is 2142, this number is a multiple of 7 BUT we want a mathematical answer.

All the results below are mathematically proved !!

This sum has 478 digits and could be superior to 4302 (9*478).

This sum is a multiple of 3 and 9.

The last digits of 3^1000 are 0001 (math proof not a result of a computer calculation).

The one who created this exercice dosen't know the answer.

Please help us with any clue! I already posted it in many forums but I still don't have any brillant idea...