> 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 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 $9$.

- The last four digits of $3^{1000}$ are $0001$.

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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 exercise doesn't know the answer.

This question was previously asked [on Math.SE (link)][1].

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