## Show that every cyclic group is abelian

In this article, we will show that every cyclic group is abelian ( or is commutative ). Basic Terminologies Abelian Group: A group that is commutative is know as abelian group. Cyclic Group: A group that contains an element g such that every other element of the group can be obtained by repeatedly applying the …

## Prove that every group of order 4 is abelian

In this article, we will prove that every group of order 4 is an abelian group. Basic Terminologies Group: A binary operation is a group if It is closed. Associative. Existence of an identity element. Each element has a unique inverse. Abelian Group: A group that is commutative is knows as abelian group, i.e., a*b …

## C Program to Read/Write Linked List to a File

In this article, we will discuss how we can read/write a linked list to a file in C programming language. Given below is the structure of the node of the linked list that we are going to use in our example Writing Linked List to a File We are using FILE object and fwrite() to …

## Keyword Cipher

Introduction Keyword Cipher is another method of encrypting alphabetic text. It is a type of monoalphabetic substitution cipher. Monoalphabetic means an alphabet in plain text is mapped to one alphabet in the ciphertext. For example, if the plain text contains multiple occurrences of ‘A’ then all occurrences of ‘A’ are substituted by the same alphabet …

## Show that n^2+n+1 is not divisible by 5

Given is not divisible by 5 Proof We will prove that the given statement is true by contradiction. Let . Assume is divisible by 5. Let, where The above equation is a quadratic equation. The roots of a quadratic equation are given by We know that . If then must also . This means that …

## Vigenere Cipher

Introduction Vigenere Cipher is a method of encrypting alphabetic text. It is a polyalphabetic substitution cipher. Polyalphabetic substitution means that an alphabet in plain text is mapped to multiple substitution alphabets. Vigenere cipher is an evolution of Caesar cipher. In Caesar cipher, we shift the alphabets of the plain text by a fixed number of …

## Prove by mathematical induction n(n+1)(2n+1) is divisible by 6

Given n(n+1)(2n+1) is divisible by 6 ∀ n ∈ N Proof STEP 1 : We will check if the statement is true for n = 1 n*(n + 1)*(2n + 1) = 1 * (1 + 1) * ( 2*1 + 1 ) = 1 * 2 * 3 = 6 The statement is true …

## Prove that if n is not divisible by 3 then n^2-1 is divisible by 3

Given If n is not divisible by 3 then n2-1 is divisible by 3 ∀ n ∈ Z Proof For the value of n, we have 3 choices Case 1 : n % 3 = 1 Let, P = (n2-1) % 3 P = ( ( n2 ) % 3 – 1 % 3 ) …

## Prove that if n is an integer then 2n^2+n+1 is not divisible by 3

Given 2n2 + n + 1 is not divisible by 3 ∀ n ∈ Z Proof We will prove the given statement by direct method. For each value n, we have three choices CASE 1 : n % 3 = 0 S = 2n2 + n + 1 Taking MOD 3 on both sides S …

## Prove that if n is an integer and 3n + 2 is odd then n is odd

Given If 3n + 2 is odd then n is odd ∀ n ∈ Z Proof By Contradiction We will prove the given statement by contradiction Suppose 3n+2 is odd. Assume n is even. If n is even, then we can write n as n = 2k, k ∈ Z Substitute n in 3n+2 3(2k) …