In this problem, we show the possible forms of any positive odd integer. We will prove that every positive odd integer can be expressed in one of the forms 6q + 1, 6q + 3, or 6q + 5, where q is an integer. Question Show that any positive odd integer is of the form6q+16q […]

In this problem, we show a property of odd positive integers. We are required to prove that the square of any odd positive integer can always be written in the form 8q + 1 for some integer q. Question Show that the square of an odd positive integer is of the form 8q + 1

In this problem, we study the form of the square of a positive integer. We are required to prove that the square of any positive integer can be written only in the forms 5q, 5q + 1, or 5q + 4 for some integer q. Question Prove that the square of any positive integer is

In this problem, we study the form of the square of a positive integer. We are required to prove that the square of any positive integer can always be written in the form 4q or 4q + 1 for some integer q. Question Prove that the square of any positive integer is of the form

In this problem, we study the form of the square of a positive integer. We are required to prove that the square of any positive integer can be written in the form 3m or 3m + 1, but never in the form 3m + 2. Question Prove that the square of any positive integer is

In this problem, we prove a property of integers expressed in algebraic form. We are required to show that the square of any positive integer of the form 5q + 1 is again of the same form. Question Prove that the square of any positive integer of the form 5q + 1 is of the

In this problem, we study the relationship between two algebraic forms of a positive integer. We are required to prove that every integer of the form 6q + 5 can also be written in the form 3q + 2 for some integer q, but the converse statement is not true. Question Prove that if a

In this problem, we prove a basic divisibility property of integers. We are required to show that for every positive integer n, the expression n³ − n is always divisible by 6. Question For any positive integer n, prove that n³ − n is divisible by 6. Solution  Consider the expressionn³ − n. Taking n

Prove That the Product of Three Consecutive Positive Integers Is Divisible by 6 Video Explanation Question Prove that the product of three consecutive positive integers is divisible by 6. Solution Step 1: Let the Three Integers Be Let the first positive integer be \(n\), where \(n\) is a positive integer. Then the next two consecutive

Prove That the Product of Two Consecutive Positive Integers Is Divisible by 2 Video Explanation Question Prove that the product of two consecutive positive integers is divisible by 2. Solution Step 1: Let the Two Consecutive Positive Integers Be Let the first positive integer be \(n\), where \(n\) is a positive integer. Then the next