Prove that the product of two integers, one of the form 3...

Prove that the product of two integers, one of the form $3 k_{1}+2$ and the other of the form $3 k_{2}+2,$ where $k_{1}$ and $k_{2}$ are integers, is of the form $3 k_{3}+1$ for some integer $k_{3}$.

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Key Concepts

Algebraic Manipulation

Algebraic manipulation is the process of using algebraic properties, such as distribution, to transform and simplify expressions. In the problem, multiplying the two numbers expressed in the form 3k + 2 and expanding the product allows one to identify and combine terms, ultimately rewriting the expression in the desired form. This systematic rearrangement shows the logical structure of the argument required in proof.

Proof by Construction

Proof by construction is a method where one directly demonstrates that a mathematical claim holds by explicitly constructing an example or expression that meets the criteria. Here, by explicitly carrying out the multiplication and rearranging terms, one constructs an expression that fits the format 3k + 1, thereby proving the claim without relying on contradiction or counterexample.

Modular Arithmetic

Modular arithmetic involves classifying integers by their remainders when divided by a fixed modulus. In this context, expressing integers in the form 3k + r (where r is the remainder when divided by 3) allows for the simplification and prediction of the results of their arithmetic operations. By recognizing numbers as equivalent to one of the residue classes modulo 3, various properties, like the behavior of products, can be systematically understood.

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