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caffeinemachine
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Let $k$ be a positive integer. Let $n=2^{k-1}$. Prove that, from $2n-1$ positive integers, one can select $n$ integers, such that their sum is divisible by $n$.
Suppose this true for some \(k\).caffeinemachine said:Let $k$ be a positive integer. Let $n=2^{k-1}$. Prove that, from $2n-1$ positive integers, one can select $n$ integers, such that their sum is divisible by $n$.
I think "equal to" is a typo. What you probably intended was "divisible by". But even that is not guaranteed (at least I am not convinced). Divisibility by $n_k$ is guaranteed though.CaptainBlack said:Suppose this true for some \(k\).
We can select \(n_k\) integers from the first set with sum divisible by \(n_k\), and a set of \(n_k\) integers from the second set with sum divisible by \(n_k\). So we have a combined set of \(2n_k=n_{k+1}\) integers from the set of $2n_{k+1}-1$ integers with sum equal to $2n_k=n_{k+1}$.
The statement means that among a set of 2n-1 integers, there exists a subgroup of n integers whose sum is divisible by n.
This statement is relevant to mathematics because it involves the concepts of divisibility, integers, and summation. It is also relevant to science because many scientific experiments involve analyzing data sets and determining if there are any patterns or relationships present.
Yes, for example, let's take the set of integers: {1, 2, 3, 4, 5}. Here, n = 3 (since 2n-1 = 5). By selecting any 3 integers from this set, we can find a subgroup whose sum is divisible by 3. For instance, {2, 3, 5} or {1, 3, 4} both have a sum of 10, which is divisible by 3.
No, this statement is not always true. It depends on the selection of integers and the value of n. For example, if we take the set of integers: {1, 2, 3, 4, 5, 6}, n = 4 and there exists no subgroup of 4 integers whose sum is divisible by 4.
This statement can be applied in various real-world situations, such as data analysis, cryptography, and statistical analysis. For example, in data analysis, this statement can help in finding patterns and relationships within a given data set. In cryptography, it can be used to determine if a set of numbers can be divided into equal parts without leaving any remainder. In statistical analysis, it can be used to test the significance of a relationship between two variables.