Divisibility Problem: Prove Existence of $i,j$

In summary, given a natural number $n$ and a sequence of integers $(a_1, \ldots, a_n)$, it is always possible to find indices $i$ and $j$ such that $i \leq j$ and the sum of elements from $i$ to $j$ in the sequence is divisible by $n$. This can be proven by considering the partial sums $b_j = \sum_{k=1}^j a_k$ and using the pigeonhole principle on the remainders when these sums are divided by $n$.
  • #1
qamaz
3
0
$$\text{ Let } n∈N \text{ and } (a1,a2,…,a_{n})∈\mathbb{Z}^{n}.

\text{ Prove that always exist } i,j∈ \underline{n} \text{ with } i≤j \text{ so }

\sum\limits_{k=i}^{\\j} a_{k} \text{ divisible by n} .$$
 
Last edited:
Physics news on Phys.org
  • #2
Hi, and welcome to the forum!

Hint: Consider \(\displaystyle b_j=\sum_{k=1}^ja_k\), $j=1,\ldots,n$ and apply the pigeonhole principle to remainders when $b_j$ are divided by $n$.
 
  • #3
What is the meaning of $$\mathbb{Z}^{n}$$?
 
  • #4
\(\displaystyle \mathbb{Z}\) denotes the set of integers. If $A$ and $B$ are sets, $A\times B$ denotes the set of all ordered pairs, where the first element comes from $A$ and the second one comes from $B$. More generally, $A_1\times \dots\times A_n$ denotes the set of all $n$-tuples, i.e., of ordered sequences of length $n$, where the $i$th element comes from $A_i$ for $i=1,\ldots,n$. Finally, $A^n=A\times\dots\times A$ ($n$ times). Thus, \(\displaystyle (a_1,\ldots,a_n)\in\mathbb{Z}^n\) means that all $a_i$ are integers.
 

FAQ: Divisibility Problem: Prove Existence of $i,j$

What is the Divisibility Problem?

The Divisibility Problem is a mathematical concept that involves determining whether one number can be divided evenly by another number. For example, the number 10 is divisible by 2 because 10/2 = 5 with no remainder. However, 10 is not divisible by 3 because 10/3 = 3 with a remainder of 1.

What is the significance of proving the existence of i and j in the Divisibility Problem?

Proving the existence of i and j in the Divisibility Problem is important because it helps to establish the divisibility relationship between two numbers. It allows us to determine if one number is a factor of another and can also be used to find common factors between two numbers.

How does one prove the existence of i and j in the Divisibility Problem?

To prove the existence of i and j in the Divisibility Problem, one must use the Division Algorithm, which states that for any two integers a and b, there exist unique integers q and r such that a = bq + r, where r is the remainder. If r is equal to 0, then b is a factor of a and i = q and j = 1. If r is not equal to 0, then b is not a factor of a and i and j do not exist.

Can i and j be negative numbers in the Divisibility Problem?

Yes, i and j can be negative numbers in the Divisibility Problem. This is because the Division Algorithm accounts for negative remainders, and thus, negative values for i and j may be necessary to prove the existence of a divisibility relationship between two numbers.

Why is proving the existence of i and j in the Divisibility Problem important in mathematics?

Proving the existence of i and j in the Divisibility Problem is important in mathematics because it allows for the simplification of equations and the identification of patterns and relationships between numbers. It also serves as a fundamental concept in higher-level mathematics, such as number theory and abstract algebra.

Similar threads

Back
Top