The Odd Numbers
An odd number is an integer that cannot be divided evenly by the number 2. Since it is not divisible by 2, it implies that any odd numbers are also not a multiple of 2.
What do you think would happen if you add +1 to any even numbers? Try to perform some calculations in your head and observe what you get. Do you obtain the same pattern when you take an even number then subtract it by 1?
Well, it’s true that an even number added by 1, or an even number subtracted by 1 will always yield an odd number.
Below are some examples to illustrate the case when an even number is increased by 1.
We can now describe the general form of an odd number. Notice, it’s very similar to the general form of an even number which is n = 2k. We simply add 1 to it. That’s why we arrive at n = 2k + 1. In fact, the remainder of an odd number when divided by 2 is always +1.
General Form of an Odd Number
DEFINITION: The number \large{n} is an odd number if it can be expressed or written as \large{2k+1} where \large{k} is just another integer.
Examples of Odd Numbers Written in General Form
Let’s put it to the test if indeed an odd number \textbf{n} can be expressed in the form \textbf{n = 2k + 1} where \textbf{k} is also an integer.
\color{red}\LARGE n = 2k + 1- - 109 \to - 109 = 2\left( { - 55} \right) + 1
- - 13 \to - 13 = 2\left( { - 7} \right) + 1
- - 5 \to - 5 = 2\left( { - 3} \right) + 1
- - 1 \to - 1 = 2\left( { - 1} \right) + 1
- 7 \to 7 = 2\left( 3 \right) + 1
- 23 \to 23 = 2\left( {11} \right) + 1
- 49 \to 49 = 2\left( {24} \right) + 1
- 101 \to 101 = 2\left( {50} \right) + 1
- 377 \to 377 = 2\left( {188} \right) + 1
- 1,013 \to 1,013 = 2\left( {506} \right) + 1
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