Prove with digit sums and the number 37

[jxrxmxn]

Hello everyone.

The task is:
,,What is the smallest Digit Sum from a positiv and natural number n ∈ ℕ, that is divideable by 37."

I have already the answer: 3
37*3 = 111 => 1+1+1=3
But now i have to prove, that 2 and 1 do not work as Digit sums.

I have already done some own work, but I have problems with proving that 2 can not be the Digit Sum of a number that is divideable by 37.

So my question is:
How can I prove, that 2 is not the smallest Digit Sum of a Number that is divideable by 37?

Here is my own part:
Numbers, of which the Digit Sum is 2, can be either displayed as a number with 2 and Zero's or as a number with two One's and Zero's.
Like 2; 11; 20; 101; 110 ; 200; 1001; 1010; 1100; ...
I have proven that 2*10^n (2; 20; 200; 2000) can not be a positiv and natural number that is divideable by 37.

I hope some of you can help me and give me a good advice! Thank you very much everyone.

 

[mmm4444bot]

… a positiv and natural number n ∈ ℕ …

It looks like you've made a nice start; you'll get some good guidance, shortly. In the meantime, here's a quick note about terminology. :cool:

Natural numbers are positive, so that phrase could infer two numbers: a Natural number and some positive number. All you need is:

n is a Natural number

or

n ∈ ℕ

 

[Dr.Peterson]

Hello everyone.

The task is:
,,What is the smallest Digit Sum from a positiv and natural number n ∈ ℕ, that is divideable by 37."

I have already the answer: 3
37*3 = 111 => 1+1+1=3
But now i have to prove, that 2 and 1 do not work as Digit sums.

I have already done some own work, but I have problems with proving that 2 can not be the Digit Sum of a number that is divideable by 37.

So my question is:
How can I prove, that 2 is not the smallest Digit Sum of a Number that is divideable by 37?

Here is my own part:
Numbers, of which the Digit Sum is 2, can be either displayed as a number with 2 and Zero's or as a number with two One's and Zero's.
Like 2; 11; 20; 101; 110 ; 200; 1001; 1010; 1100; ...
I have proven that 2*10^n (2; 20; 200; 2000) can not be a positiv and natural number that is divideable by 37.

I hope some of you can help me and give me a good advice! Thank you very much everyone.

A clearer way to ask your question would be, "What is the smallest digit sum for a positive integer n that is divisible by 37?"

Since "natural number" is defined sometimes to include zero and sometimes not, it is better to use "positive integer".

I don't yet have a good idea for how to solve it. It appears that you essentially have to prove that no number of the form 10^n + 1 is a multiple of 37. It is possible that sometime like casting out nines (or modular arithmetic more generally) might be of some use.

Where does the question come from, and what are you studying that might be useful?

 

[jxrxmxn]

Thank youu.

A clearer way to ask your question would be, "What is the smallest digit sum for a positive integer n that is divisible by 37?"

Since "natural number" is defined sometimes to include zero and sometimes not, it is better to use "positive integer".

I don't yet have a good idea for how to solve it. It appears that you essentially have to prove that no number of the form 10^n + 1 is a multiple of 37. It is possible that sometime like casting out nines (or modular arithmetic more generally) might be of some use.

Where does the question come from, and what are you studying that might be useful?

I am not studying I just love maths. I have followed your hint with modular arithmetic and found a perfect solution. Thanks