Proving properties of even, odd, and square numbers

Students will learn how certain types of numbers can be represented algebraically. First, students will learn that a number is even if and only if it can be represented as \(2k,\) where \(k\) is some integer. Likewise, a number is odd, if and only if it can be represented as \(2k + 1,\) where again, \(k\) is some integer. After that, students will learn how to prove properties involving even and odd numbers, together with addition or multiplication. See below for a list of such properties. Have students prove them until they get the gist.

• even + even = even
• even + odd = odd
• odd + odd = even
• even * even = even
• even * odd = even
• odd * odd = odd
• even^n = even, for \(n \ge 1\)
• odd^n = odd

Notice the last two properties can be proven by using induction informally. We can prove even^n = even, for \(n \ge 1,\) by using even * even = even, \(n - 1\) times. Likewise, we can prove odd^n = odd, by using odd * odd = odd, \(n - 1\) times.

We can also ask if statements involving even and odd numbers are always true, sometimes true, or never true, as seen here.

Next, students will generalize this idea of representing multiples and non-multiples, to numbers which may or may not be \(2.\) That is, even numbers are exactly the multiples of \(2,\) and odd numbers are exactly the non-multiples of \(2.\) As an example of generalizing this concept, if a whole number is one more than a multiple of \(4,\) then it can be represented as \(4n + 1,\) where \(n\) is some whole number.

After that, students will learn that a number is square, if and only if it can be represented as \(k^2,\) for some integer \(k.\) Then students will learn how to prove some basic theorems using such representations. Theorems should include those involving addition, multiplication, and squaring. For example, prove the sum of two even numbers is even. As another example, is an odd number raised to a whole number power even or odd? Make a conjecture, then prove it. These proofs require nothing more than distributing, combining like terms, and applying properties of exponents. Conclude by giving your students this challenge.