A solution to this problem?

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In summary, there are several possible solutions to replacing elements in the natural sequence such that the sum of the first k terms is divisible by k. One solution is to replace all elements (except the k:th) with 0, and let the k:th element be k (or ak for some integer a). Another solution is to replace every element from 2 onward with 1, which will result in a sum of k for the first k terms. Additionally, it is possible to double every number in the sequence, which will result in a sum of k(k+1) and is therefore divisible by k. However, this solution replaces all of the numbers in the sequence. Overall, there are various methods to achieve a sum of
  • #1
Ben1587
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Replace some elements of the natural sequence 1, 2, 3, ... such that after this replacement the sum of first k terms is divisible by k.

Any solutions?
 
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  • #2
Replace every element from 2 onward with 1. Then the sum of the first k terms will be k.
 
  • #3
Replace all elements (except the k:th) with 0. Let the k:th element be k (or ak for some integer a ;)).
 
  • #4
Ben1587 said:
Replace some elements of the natural sequence 1, 2, 3, ... such that after this replacement the sum of first k terms is divisible by k.

Any solutions?

What if by some he means 'a finite subset'?
 
  • #5
Doubling every number works.

sum(1 to k)=k(k+1)/2 so multiplying this by 2 yields k(k+1) which is of course divisible by k.

Still, that is replacing all of the numbers, but I think it is on the road to a more pleasing solution.

Njorl
 

What is a solution to this problem?

A solution to a problem is a way to resolve or fix the issue at hand. It is a method or plan of action that is designed to achieve a desired outcome or result.

Why is finding a solution important?

Finding a solution is important because it allows us to overcome challenges, improve processes, and achieve our goals. It enables us to make progress and move forward.

How do I come up with a solution?

To come up with a solution, it is important to fully understand the problem and gather all relevant information. Brainstorming, researching, and seeking input from others can also help generate ideas and potential solutions.

What are the qualities of a good solution?

A good solution should be effective in solving the problem, efficient in terms of time and resources, feasible to implement, and sustainable in the long term. It should also take into account any potential challenges or limitations.

How do I know if a solution is working?

You can measure the success of a solution by monitoring its impact and evaluating its effectiveness. This can be done through data analysis, feedback from stakeholders, and comparing the current situation to the desired outcome.

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