> Is the sum of digits of $3^{1000}$ a multiple of 7?

The sum of the digits of $3^{1000}$ can be computed using a computer. It is equal to 2142 so the answer is positive.

**Is there a short proof that the sum of the digits of $3^{1000}$ is a multiple of $7$ without using a computer?**

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

The sum calculated with Python is 2142, this number is a multiple of 7 BUT we'd like a mathematical answer (even better if applying to similar problems with other values).

All the results below are mathematically proved:

- $3^{1000}$ has 478 digits, and so the sum is at most 4302 ($9\cdot478$).

- 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).

<hr>

Context: We are a group of 3 French people working on it since 2007. It's a little exercise I found in my high school book (printed in 2007) which is pretty complicated. 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 brilliant idea...

<b>Added:</b> the question was asked [here on MathSE][1] and the reply there by "high GPA" said in particular </i>"Unexpectedly the digit sum problems are still around the hot area of research.  You could post this on the math-overflow</i>."

  [1]: https://math.stackexchange.com/questions/2433244/sum-of-digits-of-31000