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
Prove that the square of any positive integer of the form 5q + 1 is of the same form Read More »
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
For any positive integer n, prove that n³ − n is divisible by 6 Read More »
In this problem, we prove an important property of consecutive positive integers. We are required to show that the product of any three consecutive positive integers is always divisible by 6. Question Prove that the product of three consecutive positive integers is divisible by 6. Solution Let the three consecutive positive integers ben, n +
Prove that the product of three consecutive positive integers is divisible by 6 Read More »
In this problem, we prove a basic property of consecutive positive integers. We are required to show that the product of any two consecutive positive integers is always divisible by 2. Question Prove that the product of two consecutive positive integers is divisible by 2. Solution (WordPress-Safe with Basic Symbols) Let the two consecutive positive
Prove that the product of two consecutive positive integers is divisible by 2 Read More »
If a and b are two odd positive integers such that a > b, prove that one of the numbers (a + b)/2 and (a − b)/2 is odd and the other is even. In this problem, we prove that if a and b are two odd positive integers witha>ba > ba>b, then the two