Even and Odd Numbers

What are Even and Odd Numbers?

In mathematics, even numbers are integers divisible by 2; there is no remainder when the integer is divided by two. On the other hand, odd numbers are integers, when divided by 2, will have a remainder of 1. Odd numbers are always in between even numbers and vice versa. Every integer is either even or odd, and no integer is both even and odd.

Importance of Exploring Even and Odd Numbers

Determining even and odd numbers is an essential Math skill students should master. Teachers in grade levels will provide an enhanced math learning program where students learn this concept and should not just treat this lesson as a “sub-standard” lesson. Students are missing important knowledge crucial to mastering a higher degree of math concepts. Teachers should explore the deeper approaches of even and odd numbers, i.e., parity of negative integers.

Students will appreciate the lesson more if they learn it through concrete exploration. Depending on what grade level it is learned, this lesson can be taught with creative ideas. For example, instead of teaching even and odd numbers through division to first graders, it is more appropriate and creative to say that a number is even if all can be paired up without left-over. Otherwise, it’s odd. Teachers can also use visuals to present this definition.

Even and odd numbers can also be taught in higher grade levels and can have the same level of importance. In this blog from Learn ZOE, concepts such as parity and the concise definition of even and odd numbers will be delved into.

What is Parity?

The property of whether an integer is even or odd is called parity. Any two consecutive integers have opposite parity. For example, 15 is odd, and 16 is even. Even numbers can formally be defined as integers of the form n=2k, where k is an integer. It then follows that odd numbers are integers of the form n=2k+1 or n=2k-1. Note that these definitions of parity only apply to integers. Rational numbers like 25 and -5.369 and irrational numbers like π and 5 do not have parity. These concise definitions are the formulas for creating even and odd numbers and proving the parity of integers.

Example 1: Tell whether 18,596 is even or odd.

We can identify the parity of 18,596 by dividing by 2.

Since we got no remainder, the parity of 18,596 is even. We can also use the divisibility rule for 2 to identify the parity of the number. 18,596 ends with 6, so it is divisible by 2 and thus an even number.

Example 2: Identify the parity of 458,963.

Divide 458,963 by 2.

458,963 is odd since we got a remainder of 1. The number also ends with 3, an odd number, so our solution must be right.

The parity of integers also includes the following properties.

1. even even = even
2. even odd = odd
3. odd odd = even
4. even × even = even
5. even × odd = even
6. odd × odd = odd

Example 3: Tell whether 17+863 will give an even or odd sum.

Since 17 is odd, 863 is even, and the sum of an even and odd number is even, we can conclude that their sum is odd. Verify by finding the sum of the two numbers.

880 ends with zero, an even number, so the sum is indeed even.

Example 4: Determine the parity of the product of 27 and 15.

27 and 15 are odd. Therefore, their product must be odd as well. Verify by finding the product of the two numbers.

405 ends with an even number. Therefore, the product is indeed odd.

Parity of Negative Integers

Many will probably say negative integers cannot be classified as even or odd. But it turns out we actually can! The parity of negative integers seems perplexing, but it is a simple concept students need to know. To understand the concept of even and odd negative integers, we should start by generalizing the relationship of numbers in the division.

Any divisor n is equal to the sum of the product of the divisor d and quotient q, and the remainder r.

n=dq+r,

where 0r<q. This means that r cannot be negative or greater than the quotient.

This generalization can be translated to the definition of even and odd numbers. Since we determine the parity of integers by dividing by 2, d=2. The value of r can either be 0 or 1 only. 

n=2q+1 or n=2q.

We can use these two definitions to verify the parity of any negative integer.

Example 5: Prove that the parity of -37 is odd.

Let n= -37. We will use the n=2q+1 to prove its odd parity. We will show that a q exists

-37=2q+1

q=-19

Since we can show that there exists an integer q such that -37=2q+1, we see that-37 is odd. 

Word Problem Samples

Here are some examples of word problems that can be used to enhance skills in identifying even and odd numbers.

• x and y are integers such that x is even and y is odd. What is the parity of xy?

We know that even odd = even. Therefore, xy must be even.

• What is the parity of the product of the prime integers on the interval (2,1001]?

Since there are even numbers on the interval, we can say that the product of the prime numbers must also be even.

• If a is odd and b is even, which of the following must be true about (a2+b)3+ab? Select all that apply.

1. The parity of (a2+b)3+ab is even.
2. Abis even.
3. The parity of (a2+b)3+abodd.
4. (a2+b)3is even. 

Simplifying the expression using the properties of even and odd numbers, we get:

[(odd)2+even]3+(odd)(even)

  =[odd+even]+even                 

=odd+even                         

=odd                                    

Therefore, choices B and C are true.

• If x is an integer, which of the following is always odd?
  1. 2x+8
  2. 3x-5
  3. 10×2-6
  4. 34x+17

Choice A can be rewritten as 2(x+4). Multiplying by any even number will always yield an even product.

Choice B is even when x is odd.

Choice can be rewritten as 2(5×2-3). Multiplying by any even number will always yield an even product.

Choice D can be rewritten as 17(2x+1). (2x+1) defines an odd number, and 17 is odd. The product of two odd numbers is always odd. Therefore, Choice D is the correct answer.

• Prove: If x is an odd integer, then 15x+5 is even.

Any odd integer x can be written as 2n+1, where n is an integer. Therefore,

15x+2=15(2n+1)+2

          =30n+15+5 

  =30n+20

Notice that 30n+20 can be rewritten to 2(15n+10), where 15n+10 is an element of the set of integers. Since 15x+2=2(15n+10), we can conclude that 15x+2 since any integer multiplied by two yields an even integer.

Summary

• A number is even when it can be divided by 2 without a remainder. Otherwise, it is odd. When an odd is divided by 2, the remainder is always 1.
• We can formally define even numbers as 2k and odd numbers as 2k+1. We can use these two definitions to generate even and odd numbers.
• Understanding even and odd numbers is important in mastering some of the most difficult math concepts.
• Parity is the property of whether a number is even or odd. Only integers have parity.
• Negative integers also have parity. Note that the remainder, when dividing negative integers by two, must also be 0 or 1.