What makes a number odd: Odd Numbers — Definition, List of Odd Numbers
Posted onOdd Numbers — Definition, List of Odd Numbers
Odd numbers are numbers that cannot be arranged in pairs. In other words, all numbers except the multiples of 2 are odd numbers. Let us learn more about odd numbers, the definition of odd numbers, the odd numbers list and some examples of odd numbers in this article.
1.  What are Odd Numbers? 
2.  List of Odd Numbers 
3.  Properties of Odd Numbers 
4.  Types of Odd Numbers 
5.  FAQs on Odd Numbers 
What are Odd Numbers?
Odd numbers are those numbers that cannot be divided into two parts equally. In other words, odd numbers are positive integers that cannot be categorized into groups of two.
Meaning of Odd Numbers
Let us understand the meaning of odd numbers with the help of an example. If we observe numbers like, 1, 3, 5, 7, and so on, we see that these numbers cannot be paired. Let us visualize it using an example of shoes and cherries. Observe the figure given below in order to understand how the pairing of these numbers works. We have sets of shoes in counts of 1, 3, 5, and 7. On the other hand, we have cherries in pairs of 2, 4, 6, and 8.
It is to be noted here that the shoes, which show odd numbers, do not form complete pairs. One shoe among all remains unpaired. On the contrary, cherries that show even numbers are those numbers that can be divided into two parts equally and so they form complete pairs. For example, 4 cherries can be divided into 2 pairs of 2, 8 cherries can be divided into 4 pairs of 2, and so on.
Therefore, we can understand that odd numbers cannot be paired.
List of Odd Numbers
Let us have a look at the list of all the odd numbers from 1 to 200 and try to apply the knowledge we have learned here so far. Note that none of the numbers given here are multiples of 2. It should also be noted that out of the first 200 numbers, only 100 numbers are odd numbers. Have a look at the list of the odd numbers from 1 to 200 given here.
1  3  5  7  9  11  13  15  17  19 
21  23  25  27  29  31  33  35  37  39 
41  43  45  47  49  51  53  55  57  59 
61  63  65  67  69  71  73  75  77  79 
81  83  85  87  89  91  93  95  97  99 
101  103  105  107  109  111  113  115  117  119 
121  123  125  127  129  131  133  135  137  139 
141  143  145  147  149  151  153  155  157  159 
161  163  165  167  169  171  173  175  177  179 
181  183  185  187  189  191  193  195  197  199 
The definition that we have learned above is applied in this table and it eases our work, doesn’t it? Observe the given table and try to notice some similarities between all these numbers given above.
 Did you notice a pattern in the above odd numbers list?
 In the odd numbers list, the digit at the ones place always remains 1, 3, 5, 7, or 9.
Properties of Odd Numbers
The following points list the properties of odd numbers. Each of these properties can be explained in a detailed way as given below.
 Addition of Odd Numbers: The addition of two odd numbers always gives an even number, i.e., the sum of two odd numbers is always an even number. For example, 3 (odd) + 5 (odd) = 8 (even).
 Subtraction of Odd Numbers: Subtraction of two odd numbers always results in an even number. For example, 7 (odd) — 1 (odd) = 6 (even).
 Multiplication of Odd Numbers: Multiplication of two odd numbers always gives an odd number. For example, 3 (odd) × 7 (odd) = 21 (odd).
 Division Of Odd Numbers: Division of two odd numbers always gives an odd number. For example, 33 (odd) ÷ 11 (odd) = 3 (odd).
Let us summarize these properties of odd numbers using the table given below.
Arithmetic Operation  Result 

Odd number + Odd number  Even number 
Odd number — Odd number  Even number 
Odd number × Odd number  Odd number 
Odd number ÷ Odd number  Odd number 
Types of Odd Numbers
Odd numbers are a list of all the numbers that are not the multiples of 2. So this seems like a vast set of numbers. So we can have many types of odd numbers starting from whether the odd numbers have factors or not, what is the difference between two odd numbers, what is their position on the number line, etc.
Consecutive Odd Numbers
Consecutive odd numbers are those odd numbers that are listed in sequence of their order. For example, if n is an odd number, then the numbers n and n + 2 are grouped under the category of consecutive odd numbers. They always have a difference of 2 between them and are consecutive in nature, hence the name consecutive odd numbers. For example 3 and 5, 11 and 13, 25 and 27, 37 and 39, 49 and 51, and so on.
Composite Odd Numbers
Composite odd numbers are those odd numbers that are not prime numbers. These types of odd numbers are formed by the product of two smaller positive odd integers. The composite odd numbers from 1 to 100 are 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, and 99.
Smallest Odd Number
The smallest odd number is 1 because odd numbers start from 1 and are listed as, 1, 3, 5, 7, 9, 11 and so on.
First 10 Odd Numbers
The first ten odd numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19.
Smallest Odd Composite Number
The smallest odd composite number is 9. If we start looking at the smallest composite number it is 4, but that is an even number. The next composite number is 6 which is again an even number. The same goes for the next composite number which is 8. Now, the next composite number is 9 which is an odd number. So, this brings us to the smallest odd composite number which is 9.
Odd Prime Numbers
Odd prime numbers means those prime numbers which are odd numbers. Interestingly, all prime numbers except 2 are odd numbers.
Odd Natural Numbers
Odd natural numbers mean natural numbers that are odd numbers. We know that natural numbers are positive counting numbers that start from 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and go on till infinity. Among these if we select the odd numbers, we get 1, 3, 5, 7, 9, and so on.
Tips and Tricks on Odd Numbers
Given below is a list of a few tips and tricks on odd numbers. These will help you in remembering the concepts faster.
 The sum of two odd numbers is always an even number.
 An easy method to differentiate whether a number is odd or even is to divide it by 2.
 If the number is not divisible by 2 completely, it will leave a remainder of 1, which indicates that the number is an odd number and cannot be divided into 2 parts evenly.
 If the number is divisible by 2 completely, it will leave a remainder 0, which indicates that it is an even number and it can be divided into 2 parts evenly.
 Odd numbers always have 1, 3, 5, 7, or 9 in their units place. Even numbers always have 0, 2, 4, 6, or 8 in their units place.
☛ Related Articles
 Even Numbers
 Real Numbers
 Natural Numbers
 Whole Numbers
 Rational Numbers
 Irrational Numbers
 Counting Numbers
 Cardinal Numbers
 Odd Numbers 1 to 1000
 Sum of Odd Numbers
 Even and Odd Numbers Worksheets
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Examples of Odd Numbers

Example 1: Determine whether 135 is an odd number or not.
Solution:
We can find this out using any of the given two methods:
Method 1: Check the digit on the units place. The units place of the number is 5, which is an odd number. This shows that the given number is an odd number. Therefore, 135 is an odd number.
Method 2: We need to check the divisibility of the number by 2. On dividing the given number 135 by 2, we get the remainder as 1. This proves that 135 is an odd number.
Therefore, we know that the number 135 is an odd number.

Example 2: Answer the following questions with reference to odd numbers:
a. ) 1 is odd or even?
b.) Which is the smallest 4 digit odd number?
c.) What is the smallest odd composite number?
d.) What is the sum of any two odd numbers?
e.) Is 2 an odd number?
Solution:
a.) 1 is an odd number.
b.) The smallest 4 digit odd number is 1001.
c.) The smallest odd composite number is 9.
d.) The sum of any two odd numbers is always an even number.
e.) No, 2 is not an odd number, it is an even number.

Example 3: State true or false with respect to odd numbers.
a.) The sum of two odd numbers is always an even number.
b.) The smallest odd number is 5.
c.) All prime numbers are odd numbers.
d.) 9 is an odd number.
Solution:
a.) True, the sum of two odd numbers is always an even number.
b.) False, the smallest odd number is 1.
c.) False, all prime numbers are not odd numbers, because 2 is a prime number which is even. It can be said that except 2, all prime numbers are odd.
d.) True, 9 is an odd number.
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Practice Questions on Odd Numbers
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FAQs on Odd Numbers
What are Odd Numbers in Math?
In math, odd numbers are those numbers that are not the multiples of 2. For example, 3, 5, 7, 9, and so on. Odd numbers cannot be arranged in pairs which means that they cannot be divided into two parts equally.
What are the Odd Numbers From 1 to 100?
The list of odd numbers from 1 to 100 is: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99.
What is the Difference Between an Odd Number and an Even Number?
The numbers which can be divided into two parts evenly are called even numbers. For example, 4, 8, 82, and so on are even numbers. Whereas odd numbers cannot be divided into equal pairs and are not divisible by 2. For example, 7, 91, 63, and so on are odd numbers.
What are Consecutive Odd Numbers?
Two consecutive odd numbers are those numbers that are odd and have a difference of 2 between them. For example 3 and 5, 11 and 13, and so on.
Is 0 an Even or Odd Number?
0 is an even number because it is attached with the multiples of 2. For example, 10, 20, 30 40, and so on. Therefore, it is not an odd number.
What is the Mean of the First Five Odd Numbers?
In order to find the mean of the first five odd numbers we need to follow the steps listed below:
 Step 1: List the first five odd numbers, they are 1, 3, 5, 7, and 9.
 Step 2: Calculate the sum → 1 + 3 + 5 + 7 + 9 = 25.
 Step 3: Apply the mean formula = Mean of any n numbers = (Sum of n numbers) ÷ n = 25/5 = 5.
Therefore, the mean of the first five odd numbers is 5.
How Many Odd Numbers are there Between 36 and 53?
The list of odd numbers that lie between 36 and 53 are 37, 39, 41, 43, 45, 47, 49, and 51. Hence, there are a total of 8 odd numbers between 36 and 53.
Is Number 1 Odd or Even?
1 is an odd number because odd numbers are not divisible by 2 completely. For example, 3, 5, and 7 are a few odd numbers that cannot be divided equally into groups of 2..
Which Odd Number Comes After 499?
The odd number that comes after 499 is 501. The number that comes after 499 is 500 which is an even number. So, 501 is the next odd number after 499.
What is the Odd Numbers Formula?
The odd numbers formula is expressed as 2n ± 1, where, n ∈ W (whole numbers). We know that an odd number is always expressed as 1 less or more than an even number. An even number is expressed as 2n, where n is a whole number. Therefore, the formula for odd numbers is expressed as, Formula for odd numbers = 2n ± 1
What is the HCF of Two Consecutive Odd Numbers?
The Highest Common Factor (HCF) of any two consecutive odd numbers is always 1. For example, the HCF of 13 and 15 is 1.
What are Even and Odd Numbers?
All whole numbers can be categorized into two parts: even and odd numbers. Even numbers are the ones whose units place digit is either 0, 2, 4, 6, or 8, and odd numbers are ones whose units place digit is either 1, 3, 5, 7, or 9. Examples of even numbers are 2, 4, 6, 8, 10, 12, 14, 16, etc, and the examples of odd numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17, etc.
What is the Smallest Odd Number and the First Odd Number?
1 is the smallest odd number because the list of odd numbers starts from 1, 3, 5, 7 and so on. So, 1 is the smallest and the first odd number.
What are 3 Consecutive Odd Integers?
3 consecutive odd integers are those numbers that are odd and have a difference of 2 between them. For example 3 and 5, 11 and 13, and so on.
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Odd Numbers ⭐ Definition, Properties, List, Examples
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 Odd Numbers – Definition with Examples
A number is simply an arithmetic value used to represent quantity and make calculations. All numbers are formed from the numerals 09. Numbers exist as odd or even numbers and prime or composite numbers. Even numbers are all numbers that are divisible by 2 regardless of whether they are positive or negative integers. For example, 12 can be divided by 2 to give 6 without a remainder; 42 can also be divided by 2 to give 21 without a remainder.
Prime numbers are the ones that have only two prime factors, i.e. 1 and the number itself. For example, 13 has no other factors except 13 and 1. And composite numbers are opposite of prime numbers since they have more than two factors. For instance, 6 is divisible by 1, 2, 3 and 6 itself.
In this article, we are going to focus on understanding odd numbers by determining their properties, the ways to identify them, and different forms of odd numbers.
What is Odd Number?
An odd number is a number that is not exactly divisible by two. This is an inclusive odd number definition because some books define it as a number that is not divisible by two. This is not entirely true since these numbers are divisible by two, but they have remainders. Remainder is the leftover number in a division problem, and it is always less than the divisor.
For example, 19 is not exactly divisible by two 192=
9
Divisor 2 ⟌19
18
1–remainder
This makes 19 an odd number.
Odd numbers cannot be divided into two equal parts.
How to Identify Odd Numbers?
Odd numbers are identified using place values. Place value is the value of each digit in a number. The place values of numbers are read starting from right to left and are expressed as ones, tens, hundreds, thousands, and so on.
For instance, 1268.
Starting from right 8 – ones(1)
6 – tens(10)
2 – hundreds(100)
1 – thousands(1000)
For odd numbers, we focus more on one’s place value. The first value of a number from the right side is the one’s place value. If it is not exactly divisible by two, then the number is an odd number.
Here are some examples:
a) Is 15 an odd number?
The number at the one’s place value is 5. Five is not exactly divisible by 2; thus, 15 is an odd number.
b) Is 12427831 an odd number?
The numeral at the one’s place value is 1. Is one an odd number? Yes, since it cannot be exactly divided by two, the number is an odd number.
The technique of using place values is the simplest and the fastest to identify odd numbers. Otherwise, some people may use the long division method to determine if a number has a remainder or is exactly divisible by two.
Properties of Odd Numbers
Addition
When an odd number is added to another odd number, the resulting value is an even number.
Let’s take two odd numbers 3 and 9: 3+9=12, 12 is an even number.
When an odd number is added to an even number, the resulting number is an odd number. For example, 5 – odd number and 4 – even number, 4+5= 9, 9 is an odd number.
Subtraction
When an odd number is subtracted from an even number, the resulting number will be an odd number, like 103=7, 7 is an odd number.
When an odd number is subtracted from another odd number, the remainder is an even number.
53=2; so, is 2 an odd number? No, it is an even number.
When an even number is subtracted from an odd number, the remainder is an odd number, like in the examples 7 4=3, 11 6=5.
Multiplication
Multiplying an odd number with an odd number results in an odd number
7 x 3 = 21
5 x 7 = 35
Multiplying an even and odd number always results in an even number
7 x 2 = 14
6 x 5 = 30
Division
When we divide two odd numbers, the result will be an odd number. This is only true when the denominator is a factor of the numerator.
9numerator3 denominator = 3, denominator 3 is a factor of 9 where 33=9
If the denominator is an odd number but not a factor of the numerator, the resulting answer will not be a whole number but will have decimal points.
95=1,8
Types of odd numbers
There are two types of odd numbers, namely composite odd numbers and consecutive odd numbers.
Composite odd numbers
Composite numbers are positive integers with more than two factors, including 1 and the number itself. When it comes to composite odd numbers, they are positive integers which cannot be divided by 2 but have more than two factors.
For example, 9 has 3, 1, and 9 as its factors; 15 has 3,5,1, and15 as its factors.
Other examples are 9, 15, 21, 25, 27, 33, 35…
They are formed by the product of two smaller positive integers.
Consecutive odd numbers
Consecutive odd numbers as the name suggests are odd numbers that follow each other in a sequential order with a difference of two.
If x represents an odd number, then x and x+ 2 is a consecutive numbers x=3
x+2=?, 3+2=5, so, 3 and 5 are consecutive odd numbers.
The examples are 3, 5, 7, 9, 11, 13 … and so on.
The difference between the list of odd numbers given is 2; hence, these numbers are consecutive odd numbers.
Odd numbers can also be odd prime numbers or odd natural numbers.
 Odd prime numbers. These are prime numbers which are odd numbers and have only 1 and the number itself as the factors. Surprisingly, all prime numbers except 2 are odd numbers. For example, 3, 5, 7, 11,13 …
 Odd natural numbers. These are all positive counting numbers that are odd starting from 1 to infinity, such as 1, 3, 5, …
Odd number chart (1 to 100)


















































List of odd numbers from 1 to 10
1, 3, 5, 7, 9
Is zero an odd number?
Zero is not an odd number. It is actually an even number.
What is the smallest odd composite number?
From the definition of a composite odd number, 9 qualifies to be the smallest composite odd number. This is because all the other odd numbers before 9, such as 1, 3, 5, and 7, have only 1 and the number itself as the factors, while 9 has 3, 1, and 9 as the factors.
General form of odd numbers
The general form of integers is
n = 2k + 1 or n = 2k – 1, where n is the odd number and k is any positive integer.
For n = 2k + 1, the integers range from 0, 1, 2, 3 to infinity, while for n = 2k 1, the value of integers range from 0, 1, 2, 3, 4 to infinity.
Solved examples on odd numbers
Example 1
Determine if the following numbers are odd numbers:
9873457
The value at the one’s place value is 7. Since 7 is not exactly divisible by 2, then the number is an odd number.
790450
The integer at the one’s place value is 0. The number is not odd since 0 is an even number.
Example 2
Is 9 an odd number?
92= 4 remainder 1
9 is an odd number
Is 1 an odd number?
12= 0 remainder 1
1 is an odd number
Example 3
List out all the odd numbers in the following list of numbers:
1, 23, 22, 67, 32, 57, 90, 89, 64, 103, 780, 10007, 9, 33, 71
Answer: 1, 23, 67, 57, 89, 103, 10007, 9, 33, 71.
Conclusion
A number can never be an odd and even number at the same time. It can only be either even or odd. It is important to note that 1 is the first positive integer since most people rule it out and start from 3. If you add all the odd numbers from 1 to any other odd number, the resulting sum will always be a perfect square. For example, 1+ 3+ 5+ 7=16, 16 is a perfect square(4 4). To understand odd numbers, one should know its properties and types.
Frequently asked questions on odd numbers
What is an odd number?
An odd number is a number that is not exactly divisible by two.
What are the first 10 odd numbers?
They are 1, 3, 5, 7, and 9.
How do you determine if a number is odd or even?
You can determine it by dividing the number by two. If the number is exactly divisible by two, it is an even number; otherwise, it is an odd number.
What are the odd numbers from 1 to 100?
Odd numbers are all numbers with a 1, 3, 5, 7, and 9 at the one’s place value. For example, 11, 23, 35, 47, and 59.
What is the difference between an odd and even number?
An odd number is not exactly divisible by two since it has a remainder, while even numbers are exactly divisible by two. For instance, 13 and 42. 132= 6 remainder 1, then, 13 is an odd number. 422 =12, then, 42 is an even number since there is no remainder.
Video lesson in mathematics «Even and odd numbers and numbers»
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Home → Video tutorials → Mathematics. Grade 5 Integers. Description of the video lesson: Even and odd numbers are whole numbers. If a number can be divided by 2 without a remainder, then the number is even. If an integer is divided by 2 and as a result we get a number with a remainder, then the number is odd. Valery Volkov We will be glad if you share the link to this video tutorial with your friends! Education news 
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Even — odd numbers. Even and odd numbers
 Odd number
is an integer that is not divisible
without a remainder: …, 3, 1, 1, 3, 5, 7, 9, …
if m
is even, then it can be represented as m = 2 k, and if odd, then as m = 2 k + 1, where k \in \mathbb Z.
History and culture
The concept of parity of numbers has been known since ancient times and has often been given a mystical meaning. In Chinese cosmology and natural philosophy, even numbers correspond to the concept of «yin», and odd — «yang».
In different countries there are traditions related to the number of flowers given. For example, in the USA, Europe and some eastern countries, it is believed that an even number of flowers given brings happiness. In Russia and the CIS countries, it is customary to bring an even number of flowers only to the funerals of the dead. However, in cases where there are many flowers in the bouquet (usually more), the evenness or oddness of their number no longer plays any role. For example, it is quite acceptable to give a lady a bouquet of 12, 14, 16, etc. flowers or sections of a spray flower that have many buds, in which they, in principle, are not counted. This applies even more to the larger number of flowers (cuts) given on other occasions.
Practice
Even and odd weeks are used in higher education institutions with complex study schedules. Within these weeks, the schedule of training sessions and, in some cases, their start and end times differ. This practice is used to evenly distribute the load across classrooms, educational buildings and for the rhythm of classes in disciplines with a small classroom load (1 time in 2 weeks)
In train schedules, even and odd train numbers are used, depending on the direction of movement ). Accordingly, even/odd indicates the direction in which the train passes through each station.
Even and odd days of the month are sometimes associated with train schedules, which are organized every other day.
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Notes
Links
 OEIS sequence A005408 : odd numbers
 OEIS sequence A005843 : even numbers
 OEIS sequence A179082 : even numbers with an even sum of digits in decimal notation
An excerpt characterizing even and odd numbers
— So, so, — said Prince Andrei, turning to Alpatych, — tell everything as I told you. And, without answering a word to Berg, who fell silent beside him, he touched the horse and rode into the alley.
The troops continued to retreat from Smolensk. The enemy was following them. On August 10, the regiment, commanded by Prince Andrei, passed along the high road, past the avenue leading to the Bald Mountains. The heat and drought lasted for more than three weeks. Curly clouds moved across the sky every day, occasionally obscuring the sun; but towards evening it cleared again, and the sun set in a brownishred mist. Only heavy dew at night refreshed the earth. The bread remaining on the root burned and spilled out. The swamps have dried up. The cattle roared from hunger, not finding food in the meadows burned by the sun. Only at night and in the forests the dew still held, it was cool. But along the road, along the high road along which the troops marched, even at night, even through the forests, there was no such coolness. The dew was not noticeable on the sandy dust of the road, which was pushed up more than a quarter of an arshin. As soon as it dawned, the movement began. Convoys, artillery silently walked along the hub, and the infantry up to their ankles in soft, stuffy, hot dust that had not cooled down during the night. One part of this sandy dust was kneaded by feet and wheels, the other rose and stood like a cloud over the army, sticking to the eyes, hair, ears, nostrils and, most importantly, the lungs of people and animals moving along this road. The higher the sun rose, the higher the cloud of dust rose, and through this thin, hot dust it was possible to look at the sun, not covered by clouds, with a simple eye. The sun was a big crimson ball. There was no wind, and people were suffocating in this still atmosphere. People walked with handkerchiefs around their noses and mouths. Coming to the village, everything rushed to the wells. They fought for water and drank it to the dirt.
Prince Andrei commanded a regiment, and the organization of the regiment, the wellbeing of his people, the need to receive and give orders occupied him. The fire of Smolensk and its abandonment were an epoch for Prince Andrei. A new feeling of bitterness against the enemy made him forget his grief. He was completely devoted to the affairs of his regiment, he was caring for his people and officers and affectionate with them. In the regiment they called him our prince, they were proud of him and loved him. But he was kind and meek only with his regimental officers, with Timokhin, etc. , with completely new people and in a foreign environment, with people who could not know and understand his past; but as soon as he ran into one of his former staff members, he immediately bristled again; became malicious, mocking and contemptuous. Everything that connected his memory with the past repulsed him, and therefore he tried in the relations of this former world only not to be unjust and to fulfill his duty.
True, everything was presented in a dark, gloomy light to Prince Andrei — especially after they left Smolensk (which, according to his concepts, could and should have been defended) on August 6, and after his father, who was sick, had to flee to Moscow and throw away the so beloved, built and populated by him Bald Mountains for plunder; but, despite the fact, thanks to the regiment, Prince Andrei could think about another subject, completely independent of general questions — about his regiment. On August 10, the column, in which his regiment was, caught up with the Bald Mountains. Prince Andrey two days ago received the news that his father, son and sister had left for Moscow. Although Prince Andrei had nothing to do in the Bald Mountains, he, with his characteristic desire to inflame his grief, decided that he should call in the Bald Mountains.
He ordered his horse to be saddled and rode from the crossing to his father’s village, where he was born and spent his childhood. Passing by a pond, where dozens of women, talking to each other, beat with rollers and rinsed their clothes, Prince Andrei noticed that there was no one on the pond, and a tornoff raft, half flooded with water, floated sideways in the middle of the pond. Prince Andrei drove up to the gatehouse. There was no one at the stone entrance gate, and the door was unlocked. The garden paths were already overgrown, and the calves and horses were walking through the English park. Prince Andrei drove up to the greenhouse; the windows were broken, and the trees in tubs, some felled, some withered. He called Taras the gardener. Nobody responded. Going around the greenhouse to the exhibition, he saw that the carved board fence was all broken and the plum fruits were plucked with branches. An old peasant (Prince Andrei had seen him at the gate in his childhood) was sitting and weaving bast shoes on a green bench.
He was deaf and did not hear Prince Andrei’s entrance. He was sitting on a bench, on which the old prince liked to sit, and beside him was hung a bast on the knots of a broken and withered magnolia.
Prince Andrei drove up to the house. Several lindens in the old garden were cut down, one piebald horse with a foal walked in front of the house between the roses. The house was boarded up with shutters. One window downstairs was open. The yard boy, seeing Prince Andrei, ran into the house.
Alpatych, having sent his family, remained alone in the Bald Mountains; he sat at home and read the Lives. Upon learning of the arrival of Prince Andrei, he, with glasses on his nose, buttoning up, left the house, hurriedly approached the prince and, without saying anything, wept, kissing Prince Andrei on the knee.
What do even numbers mean?
and odd numbers
in spiritual numerology. This is a very important topic in the study! What is the difference between even numbers and odd numbers?
Even numbers
It is common knowledge that even numbers
those that are divisible by two. That is, the numbers 2, 4, 6, 8, 10, 12, 14, 16, 18 and so on.
What do even numbers mean?
relatively ? What is the numerological essence of dividing by two? And the bottom line is that all numbers that are divisible by two carry some of the properties of two.
Multiple values. Firstly, this is the most «human» figure in numerology. That is, the number 2 reflects the whole gamut of human weaknesses, shortcomings and virtues — more precisely, what is considered in society to be virtues and shortcomings, «correctness» and «incorrectness».
And since these labels of “correctness” and “incorrectness” reflect our limited views of the world, then the deuce can be considered the most limited, most “stupid” number in numerology. From this it is clear that even numbers are much more “hardheaded” and straightforward than their odd counterparts, which are not divisible by two.
This, however, does not mean that even numbers are worse than odd numbers. They are just different and reflect other forms of human existence and consciousness in comparison with odd numbers. Even numbers in spiritual numerology always obey the laws of ordinary, material, «earthly» logic. Why?
Because another meaning of two: standard logical thinking. And all even numbers in spiritual numerology, one way or another, obey certain logical rules for the perception of reality.
An elementary example: if a stone is tossed up, it, having gained a certain height, will then rush towards the ground. This is how even numbers «think». And odd numbers will easily assume that the stone will fly into space; or not fly, but get stuck somewhere in the air … for a long time, for centuries. Or just dissolve! The more illogical the hypothesis, the closer it is to odd numbers.
Odd numbers
Odd numbers
— those that are not divisible by two: the numbers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21 and so on. From the standpoint of spiritual numerology, odd numbers are subject not to material, but to spiritual logic.
Which, by the way, gives food for thought: why the number of flowers in a bouquet is odd for a living person, and even for a dead person… Is it because the material logic (logic within the framework of “yesno”) is dead in relation to the human soul?
Visible coincidences of material logic and spiritual occur very often. But don’t let that fool you. The logic of the spirit, that is, the logic of odd numbers
, is never fully traced at the external, physical levels of human existence and consciousness.
Take, for example, the number of love. We talk about love at every turn. We confess it, dream about it, decorate our lives and other people’s lives with it.
But what do we really know about love? About that allpenetrating Love that permeates all spheres of the Universe. Can we agree and accept that there is as much cold in it as warmth, as much hatred as kindness?! Are we able to realize that it is these paradoxes that make up the highest, creative essence of Love?!
Paradoxicality is one of the key properties of odd numbers. In the interpretation of odd numbers
It must be understood that what seems to a person is not always really existing. But at the same time, if something seems to someone, then it already exists. There are different levels of Existence, and illusion is one of them…
By the way, the maturity of the mind is characterized by the ability to perceive paradoxes. Therefore, it takes a little more «brains» to explain odd numbers than to explain even numbers.
Even and odd numbers in numerology
Let’s summarize. What is the main difference between even and odd numbers?
Even numbers are more predictable (except for the number 10), solid and consistent. Events and people associated with even numbers are more stable and explainable. Quite accessible for external changes, but only for external ones! Internal changes are the area of odd numbers…
Odd numbers are eccentric, freedomloving, unstable, unpredictable. They always bring surprises. It seems that you know the meaning of some odd number, and it, this number, suddenly begins to behave in such a way that it makes you reconsider almost your entire life …
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So I’ll start my story with even numbers. What are even numbers? Any integer that can be divided by two without a remainder is considered even. In addition, even numbers end with one of the given number: 0, 2, 4, 6 or 8.
For example: 24, 0, 6, 38 are all even numbers.
m = 2k is the general formula for writing even numbers, where k is an integer. This formula may be needed to solve many problems or equations in elementary grades.
There is another kind of numbers in the vast realm of mathematics — these are odd numbers. Any number that cannot be divided by two without a remainder, and when divided by two, the remainder is equal to one, is called odd. Any of them ends with one of these numbers: 1, 3, 5, 7 or 9.
An example of odd numbers: 3, 1, 7 and 35.
n = 2k + 1 is a formula that can be used to write any odd numbers, where k is an integer.
Addition and subtraction of even and odd numbers
There is a pattern in adding (or subtracting) even and odd numbers. We have presented it with the help of the table below, in order to make it easier for you to understand and remember the material.
Operation 
Result 
Example 
Even + Even 

Even + Odd 
Odd 

Odd + Odd 
Even and odd numbers will behave the same way if you subtract rather than add them.
Multiplication of even and odd numbers
Even and odd numbers behave naturally when multiplied. You will know in advance whether the result will be even or odd. The table below shows all possible options for better assimilation of information.
Operation 
Result 
Example 
Even * Even 

Even * Odd 

Odd * Odd 
Odd 
And now consider fractional numbers.
Decimal representation of a number
Decimal fractions are numbers with a denominator of 10, 100, 1000, and so on, which are written without a denominator. The integer part is separated from the fractional part with a comma.
For example: 3.14; 5.1; 6.789 is everything
You can perform various mathematical operations with decimals, such as comparison, summation, subtraction, multiplication, and division.
If you want to compare two fractions, first equalize the number of decimal places by assigning zeros to one of them, and then, discarding the comma, compare them as whole numbers. Let’s look at this with an example. Let’s compare 5.15 and 5.1. First, let’s equalize the fractions: 5.15 and 5.10. Now we write them as integers: 515 and 510, therefore, the first number is greater than the second, so 5.15 is greater than 5.1.
If you want to add two fractions, follow this simple rule: start at the end of the fraction and add first (for example) hundredths, then tenths, then integers. With this rule, you can easily subtract and multiply decimal fractions.
But you need to divide fractions as whole numbers, counting at the end where you need to put a comma. That is, first divide the whole part, and then the fractional part.
Decimal fractions should also be rounded. To do this, select to what decimal place you want to round the fraction, and replace the corresponding number of digits with zeros. Keep in mind if the number following this digit was in the range from 5 to 9inclusive, then the last digit that remains is increased by one. If the digit following this digit lay in the range from 1 to 4 inclusive, then the last remaining one does not change.
Definitions
 Even number
is an integer which is divisible by
no remainder on 2: …, −4, −2, 0, 2, 4, 6, 8, …  Odd number
is an integer that is not divisible
no remainder on 2: …, 3, 1, 1, 3, 5, 7, 9, …
According to this definition, zero is an even number.
if m
is even, then it can be represented as , and if odd, then as , where .
In different countries there are traditions related to the number of flowers given.
In Russia and the CIS countries, it is customary to bring an even number of flowers only to the funerals of the dead. However, in cases where there are many flowers in the bouquet (usually more), the evenness or oddness of their number no longer plays any role.
For example, it is perfectly acceptable to give a young lady a bouquet of 12 or 14 flowers or sections of a bush flower, if they have many buds, in which they, in principle, are not counted.
This applies even more to the larger number of flowers (cuts) given on other occasions.
Notes
Wikimedia Foundation
.
2010
.
 Maardu
 Superconductivity
See what «Even and odd numbers» are in other dictionaries:
Odd numbers
Even numbers
— Parity in number theory is a characteristic of an integer, which determines its ability to be divided by two. If an integer is divisible by two without a remainder, it is called even (examples: 2, 28, −8, 40), if not odd (examples: 1, 3, 75, −19). … … Wikipedia
Odd
— Parity in number theory is a characteristic of an integer, which determines its ability to be divided by two. If an integer is divisible by two without a remainder, it is called even (examples: 2, 28, −8, 40), if not odd (examples: 1, 3, 75, −19).… … Wikipedia
Odd number
— Parity in number theory is a characteristic of an integer, which determines its ability to be divided by two. If an integer is divisible by two without a remainder, it is called even (examples: 2, 28, −8, 40), if not odd (examples: 1, 3, 75, −19). … … Wikipedia
Odd numbers
— Parity in number theory is a characteristic of an integer, which determines its ability to be divided by two. If an integer is divisible by two without a remainder, it is called even (examples: 2, 28, −8, 40), if not odd (examples: 1, 3, 75, −19).… … Wikipedia
Even and odd numbers
— Parity in number theory is a characteristic of an integer, which determines its ability to be divided by two. If an integer is divisible by two without a remainder, it is called even (examples: 2, 28, −8, 40), if not odd (examples: 1, 3, 75, −19). … … Wikipedia
Even numbers
— Parity in number theory is a characteristic of an integer, which determines its ability to be divided by two. If an integer is divisible by two without a remainder, it is called even (examples: 2, 28, −8, 40), if not odd (examples: 1, 3, 75, −19).… … Wikipedia
Slightly redundant numbers
— A slightly redundant number, or a quasiperfect number is an excess number whose sum of its own divisors is one more than the number itself. So far, no slightly redundant numbers have been found. But since the time of Pythagoras, … … Wikipedia
Perfect numbers
— positive integers equal to the sum of all their correct (i.e. less than this number) divisors. For example, the numbers 6 = 1+2+3 and 28 = 1+2+4+7+14 are perfect. Even Euclid (3rd century BC) indicated that even S. hours can be … …
Quantum numbers
— integer (0, 1, 2,…) or halfinteger (1/2, 3/2, 5/2,…) numbers that determine the possible discrete values of physical quantities that characterize quantum systems (atomic nucleus, atom, molecule) and individual elementary particles. … … Great Soviet Encyclopedia
Books
 Mathematical labyrinths and puzzles, 20 cards, Barchan Tatyana Alexandrovna, Samodelko Anna. In the set: 10 puzzles and 10 mathematical labyrinths on the topics: — Numerical series; — Even and odd numbers; — Composition of the number; — Counting in pairs; — Exercises for addition and subtraction. Includes 20…
There are pairs of opposites in the universe, which are an important factor in its structure. The main properties that numerologists attribute to even (1, 3, 5, 7, 9) and odd (2, 4, 6, 8) numbers, as pairs of opposites, are as follows:
1 — active, purposeful, imperious, callous, leading, enterprising;
2 — passive, receptive, weak, sympathetic, subordinate;
3 — bright, cheerful, artistic, lucky, easily achieving success;
4 — hardworking, boring, lack of initiative, unhappy, hard work and frequent defeat;
5 — mobile, adventurous, nervous, insecure, sexual;
6 — simple, calm, homely, arranged; mother’s love;
7 — withdrawal from the world, mysticism, secrets;
8 — worldly life; material success or failure;
9 — intellectual and spiritual perfection.
Odd numbers have much brighter properties. Next to energy «1», brilliance and luck «3», adventurous mobility and versatility «5», wisdom «7» and perfection «9»Even numbers do not look so bright. There are 10 main pairs of opposites that exist in the Universe. Among these pairs: even — odd, one — many, right — left, male — female, good — evil. One, right, male and good were associated with odd numbers, many, left, feminine and evil — with even
Odd numbers have a certain generating middle, while in any even number there is a perceiving hole, as it were, a gap inside itself.Male properties of phallic odd numbers follow from the fact that they are stronger than even ones.If you split an even number in half, then there will be nothing left except for emptiness in the middle.An odd number is not easy to split, because a dot remains in the middle.If you put an even and an odd number together, then the odd one wins, since the result is always will be odd, which is why odd numbers are masculine, powerful and harsh, while even numbers are feminine, passive and receptive. 0005
Odd numbers odd number: there are five. Even numbers an even number — four.
Odd numbers — solar, electrical, acidic and dynamic. They are terms; stack them with something. Even numbers are lunar, magnetic, alkaline and static. They are deductible, they are reduced. They remain motionless because they have even groups of pairs (2 and 4; 6 and 8).
If we group odd numbers, one number will always be left without its pair (1 and 3; 5 and 7; 9). This makes them dynamic. Two similar numbers (two odd numbers or two even numbers) are not auspicious.
even + even = even (static) 2+2=4
even + odd = odd (dynamic) 3+2=5
odd + odd = even (static) 3+3=6
Some numbers are friendly, others — oppose each other. The relationship of numbers is determined by the relationship between the planets that rule them (details in the «Number Compatibility» section). When two friendly numbers touch, their cooperation is not very productive. Like friends, they relax — and nothing happens.