Prove that, for any k ∈N a t, 3 |k(k+1)(k+2). [Hint. By the Division Algorithm Theorem, k=3 q+r,

step 1 All right, so you want to show three divides n cubed minus 3. So we'll consider all the integers

Prove that, for any k ∈N a t, 3 |k(k+1)(k+2). [Hint. By the...

Key Concepts

Division Algorithm

The Division Algorithm is a fundamental concept in number theory stating that any integer can be expressed in the form k = dq + r, where d is a positive integer (the divisor), q is the quotient, and r is the remainder satisfying 0 ? r < d. This concept is crucial when analyzing divisibility properties, as it allows one to break down any integer into components that simplify divisibility arguments.

Modular Arithmetic

Modular arithmetic is the study of integers under a modulus, where numbers are considered equivalent if they yield the same remainder when divided by a fixed modulus. This concept is essential in proofs of divisibility because it provides a systematic way to handle remainders and shows that properties like divisibility are preserved under the arithmetic operations within specific residue classes.

Proof by Cases

Proof by cases is a logical methodology where a proposition is established by demonstrating it holds under several distinct scenarios. In the context of divisibility proofs, different cases based on the possible remainders from the Division Algorithm are considered, ensuring that all potential instances are covered and that the conclusion follows universally.

Product of Consecutive Integers

The product of consecutive integers exhibits regular divisibility properties because within any set of consecutive numbers, the distribution of remainders ensures that at least one of the numbers is divisible by any small prime such as 3. This fact is often used to establish general divisibility rules in number theory by leveraging the regular spacing of multiples in the sequence of integers.

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