Proving the Product of Any Two Even Integers is Even

To prove that the product of any two even integers is even, we can follow a structured proof that leverages the basic definition of even integers. Understanding this concept is fundamental in number theory and essential for various mathematical proofs.

Definition of Even Integers

An integer n is defined as even if it can be expressed in the form n 2k, where k is an integer. This definition is the cornerstone of our proof.

Setting Up the Integers

Let’s define two even integers:

The first even integer is a 2m, for some integer m.

The second even integer is b 2n, for some integer n.

Calculating the Product

We need to calculate the product of these two even integers:

a cdot b  2m cdot 2n

Simplifying the Product

Simplifying the above expression, we get:

a cdot b  4mn

Rearranging the Expression

Rearranging the expression, we can write:

4mn  2 cdot 2mn

Here, 2mn is an integer since the product of two integers m and n is also an integer.

Conclusion

Since a cdot b can be expressed in the form 2k where k 2mn, we conclude that a cdot b is even. This proof demonstrates that the product of any two even integers is also an even integer.

Extending the Proof to Multiples of 4

Moreover, we can extend this proof to show that the product of any two even integers is not only even but also a multiple of 4.

Proof that the Product is a Multiple of 4

Let m and n be integers.

Two even integers are 2m and 2n.

The product of these two even integers is 2m times; 2n 4mn.

Let p mn. Since m and n are both integers, p must also be an integer. Therefore, 4mn 4p.

Since 4p is a multiple of 4, we conclude that the product of any two even integers is both even and a multiple of 4.

Final Steps for a Deductive Proof

Let's restate each of the steps to provide a complete deductive proof:

An integer n is called even if and only if there exists an integer k such that n 2k.

Premise: Assume that a and b are both even integers.

Then there exist integers i and j such that a 2i and b 2j.

Note that ab 2i middot; 2j 4ij 2 middot; 2ij.

Since 2ij is an integer, it follows that ab is even.

This proof clearly demonstrates the properties of even integers and their arithmetic operations. Understanding these properties is crucial for various mathematical applications and proofs.