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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 $0001$.

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).

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 known:

• $3^{1000}$ 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 $0001$.

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).

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!)?

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).

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 here on Math.SE.

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 $0001$.

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).

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).

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...

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

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!)?

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).

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 here on Math.SE.