Educational

A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, 3m or 3m + 2? Justify your answer. Introduction In this problem, we examine the square of a positive integer of the form 3q + 1. We […]

Show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q Introduction In this problem, we study the form of the square of an odd positive integer. We will show that the square of an odd positive integer can be written

Show that one and only one out of n, n+4, n+8, n+12 and n+16 is divisible by 5, where n is any positive integer Introduction In this problem, we study the divisibility of a set of five numbers. We will show that among the numbers n, n+4, n+8, n+12 and n+16, exactly one number is

Show that the cube of a positive integer is of the form 6q + r, where q is an integer and r = 0, 1, 2, 3, 4, 5 Introduction In this problem, we study the possible forms of the cube of a positive integer. We will show that the cube of any positive integer

Show that the square of any positive integer cannot be of the form 6q + 2 or 6q + 5 for any integer q Introduction In this problem, we study the possible forms of the square of a positive integer. We will show that the square of any positive integer can never be written in

In this problem, we prove a basic property of consecutive positive integers. We will show that among any three consecutive positive integers, one integer is always divisible by 3. Question Prove that one of every three consecutive positive integers is divisible by 3. Solution Let the three consecutive positive integers be n,  n+1,  n+2n,\; n+1,\; n+2n,n+1,n+2 where

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