Even Numbers 

The fundamental units of mathematics are numbers. They aid in problem-solving, measurement, comparison, and counting. These figures are divided into various categories. The division of numbers into even numbers is one of the most basic and straightforward classifications among these various number types, and it is the subject of this article. So let's get started!

1.0Even Numbers Definition

A number that is a multiple of two is called an even number. Any number that is totally divisible by two can be considered even. An example can help you understand this.

In essence, all numbers—positive, negative, or zero—can be divided into two equal groups with no leftovers. For example, 6 is an even number since it can be divided into two groups of three. Because 10 can be divided equally into 5 and 5, it is an even number. Hence, the definition of even numbers is based on the concept of perfect grouping.

This can all be written as:  n=2k

Here n is an even number and k is an integer. On the basis of this formula, some even numbers include:  

• 2 × 0 = 0 → 0 is even.
• 2 × 5 = 10 → 10 is even.
• 2 × (−4) = −8 → –8 is even

2.0Even Numbers Properties

1. Divisibility by 2: This is what is meant by the fact that each number's ability to be divided by 2 without a remainder explains that each number is even. For instance:

• 26 ÷ 2  = 13 → is a whole number → 26 is even.
• −14 ÷ 2 = −7 → is a whole number → –14 is even.

2. Adding Even Numbers: 

• When adding any two even numbers, the result will always be an even number. For instance:

1. 18 + 20 = 38 (even)
2. −4 + −6 = −10 (even)

• Any even number added to any odd number will always result in an odd number; e.g.:

1. 15+12=27 (odd)
2. –6+(–5)=–11 (odd)
3. Subtracting Even Numbers: 

• Any two even numbers' differences will always be even. For example: 

1. 40−12=28 (even) 
2. −8−(−4)=−4  (even)

• The difference of any even number from an odd number will always be equal to an odd number; for example: 

1. 58−43=15 (odd) 
2. −85−74=−159 (odd) 

4. Multiplying With Integers: The product of any integer (odd or even) and an even number will always be even. Such as: 

• 6×3=18
• 12×12=144

5. Last digit property: Any even number will always end in either 0, 2, 4, 6, or 8. For example: 

• 342 → ends in 2 → even 
• 5698 →  ends in 8 → even 

Note: This is the best way to recognise an even number without needing to divide it by 2. 

6. Successive Even Numbers: Any two consecutive even numbers always differ by 2. And the general form for these even numbers is 2n, 2n+2, 2n+4, and so on. Examples for two successive even numbers include: 

• 10 and 12.
• –6 and –4.

7. Squaring Property: The square of any even number is always equal to an even number. For instance: 

• 8² = 64.
• −12² = 144.

8. Powers of Even Numbers: Any power of an even number will always result in an even number. For example: 

• 2⁵ = 32
• –6³ = –216

3.0Difference Between Even and Odd Numbers: 

4.0Sum of "n" Even Numbers

The sum of n even numbers is an important formula to find the sum of a certain number of even numbers in a given series. The formula is derived from the general formula of the sum of n terms in an arithmetic progression by taking two as the common difference. That is: 

The general formula of the sum of n terms of an arithmetic progression: 

${\mathrm{S}}_{\mathrm{n}}=\frac{n}{2}(2a+(n-1)d)$

For even numbers, the common difference(d) will be equal to 2. Therefore, the new formula will be: 

$\begin{array}{c}{\mathrm{S}}_{\mathrm{n}}=\frac{n}{2}(2a+(n-1)2)\\ {\mathrm{S}}_{\mathrm{n}}=\frac{n}{2}(2a+2n-2)\\ {\mathrm{S}}_{\mathrm{n}}=n(a+n-1)\end{array}$

Now, for the sum of the first n even numbers, the first term or a will be 2, so the formula will again be modified as: 

$\begin{array}{c}{\mathrm{S}}_{\mathrm{n}}=n(2+n-1)\\ {\mathrm{S}}_{\mathrm{n}}=n(n+1)\end{array}$

Understand this with this example: Find the sum of the first 50 even numbers.

Here, the sum of the first 50 even numbers will be S₅₀: 

$\begin{array}{rl}& S_{50}=50(50+1)=50\times 51\\ & S_n=2550\end{array}$