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
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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
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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
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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 »