Study of even and odd numbers. Even and odd numbers: what do they mean in numerology

Shnyakina Alina

The work took II place at the regional scientific - practical conference.

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Regional scientific - practical conference students and teachers

Subject "mathematics"

Nomination " Abstract of a problem-search nature»

Topic: "Even and Odd Numbers"

Teacher: Okshina L.A.

2011 - 2012 academic year

MOBU "Rybkinskaya secondary school"

Even and odd numbers.

/Abstract/

Work completed

5th grade student

Shnyakina Alina.

Work checked

Mathematic teacher

Okshina L. A.

With. Rybkino 2012

Introduction 4

Main part 5

Definition. Properties. 5

Traditions 6

Pythagorean number theory 8

Numerology 10

Conclusion 12

Literature 13

Introduction.

Purpose: to find out why odd and even numbers are assigned different meanings.

Tasks:

1. Find the definition and properties of even and odd numbers.
2. What traditions in different countries are associated with numbers?
3. How are even and odd numbers used in numerology?

Plan:

1. Introduction.
2. Main part.

1. Definition. Properties;

1. Traditions;
2. Pythagorean number theory;
3. Numerology.

1. Conclusion.

Relevance.

Even in ancient times, people noted the influence of numbers and the dependence of fate on the coincidence or, conversely, non-falling out of certain numbers, as well as the cyclical nature of everything that happens in the world. Not philosophers and not thinkers, most likely, in general, in their mass, simple and not very educated people very correctly expressed this in fairy tales and myths, where three and seven most often appear.

In fairy tales, there were three heroes, three or seven sons, seven gnomes, and the kingdom was considered far away! In order not to jinx their luck, people traditionally spat three times (and now too!) Over the left shoulder or knocked on wood. Especially favorite numbers are often found in proverbs and sayings: “God loves a trinity”, “seven do not wait for one”, “measure seven times, cut once” ...

Why are odd numbers mostly used in fairy tales?

Why do people give an odd number of flowers for a birthday? And there were many more questions in front of me.

I decided to find out about it. I found the material and began my research.

Main part.

Definition.

1. An even number is an integer that is divided no remainder by 2: for example: 2, 4, 6, 8, ...
2. An odd number is an integer that not shared no remainder by 2: for example: 1, 3, 5, 7, 9, ...

According to this definitionzerois an even number.

If in decimal notation of a number last digit is an even number (0, 2, 4, 6 or 8), then the whole number is also even, otherwise it is odd.

42, 104, 11110, 9115817342 are even numbers.

31, 703, 78527, 2356895125 - odd numbers

Properties.

1. Division:

1. Even / Even - it is impossible to unambiguously judge the parity of the result (if the resultinteger , it can be either even or odd)
2. Even / Odd = if resultinteger, then it is Even
3. Odd / Even - the result cannot be an integer, and therefore have parity attributes
4. Odd / Odd = if resultinteger , then it is Odd.

Traditions.

The concept of parity of numbers has been known since ancient times and was often given to it. mystical meaning. IN different countries there are traditions associated with the number of donationscolors.

For example, in Europe, the USA and some countries of the East, there is a belief that an even number of flowers brings happiness.

According to Russian traditions, an even number of flowers is brought to a funeral, and it is customary for a living person to give flowers only in an odd number.
There are several versions about the origin of this tradition.
pagan beliefs interpret even numbers as symbols of death and evil. Remember the saying "trouble never comes alone"? It was from following this tradition that the custom came to give living people only an even number of flowers.

Many ancient cultures associated paired numbers with completeness, completion, in this case, the life path. An odd number, (except 13), on the contrary, is a symbol of happiness, success, luck. Odd numbers are unstable, they symbolize movement, life, laughter. Even - a symbol of peace and tranquility.

For the ancient Pythagoreans, odd numbers were the personification of goodness, life, light, and they also symbolized right side(side of luck). The unfortunate left side, and with it death, evil, darkness, was symbolized by even numbers.

Isn't that where the famous "stand on your left foot" came from, symbolizing a bad start to the day? IN Japanese culture the numbers 1,3,5 stand for masculinity"yang" and speak of life, strength, movement. The numbers 2,4,6 are feminine"yin", peace, passivity. In Japan, it is not customary to give living people four flowers, because the number 4 symbolizes death.

The Israelis, on the contrary, give an even number of flowers, but they don’t bring flowers to funerals. Georgia believes that everything related to family values brings happiness, so two flowers (pair) - good combination, and an odd number of flowers are carried to the cemetery "so that the deceased does not take the couple with him." A European or an American can, with the best of intentions, give a Russian girl 8 or 10 roses and be genuinely surprised by her reaction.

It is worth noting that such a picky count of flowers takes place only up to a dozen. After this amount, it does not matter if the number of stems in the bouquet is even or not. After all, the notorious "million scarlet roses" - has an even number of flowers.

In many fairy tales we are meeting different numbers. Most often these are the numbers THREE and SEVEN.

The number "3" has long been considered magical. Even in the Bible, God appears in a triune person. 3 is divine perfection. There is an expression: God loves a trinity.

The number "3" in fairy tales makes the reader think about magic, about perfection. Indeed, in Russian fairy tales, desires are always fulfilled only for the third time.

"Three maidens by the window

Spinning late at night."

"And they will find themselves on the shore,

In scales like the heat of grief,

Thirty-three heroes."

7 is a special number. So, it is known that the priests of Babylon worshiped seven gods. The symbolism of the number 7 is also characteristic of biblical stories. Theologians interpret this number as a combination of the number of 3-divine perfection and 4-world order.

In Russian sayings and proverbs, the word “seven” often appears in the meaning of “many”: “Seven do not expect one”, “Measure seven times - cut once”, “Seven troubles - one answer”, “Onion from seven ailments”, etc. d. In the tale of A.S. Pushkin's number 7 also means "a lot": "seven heroes, seven ruddy mustaches."

As a dowry, the princess was given "seven trading cities and one hundred and forty towers."

But with even numbers there is a superstitious idea: it is associated with death, with evil spirits.

This means that the choice of numerals in fairy tales is based on the popular idea of ​​the meaning of numbers.

Pythagorean number theory.

Is it possible to know how many joys happy days, troubles and misfortunes are destined in life for each of us? In search of an answer, people have long, according to their observations, began to attribute a special magical meaning to numbers. This made it possible to interpret the dependence of phenomena on numbers and to explain their laws. Thus was born the science of numbers - numerology. A special role in the development of numerology belongs to the great Pythagoras, the ancient Greek philosopher and mathematician, who combined mathematics with the sciences of human nature.

Numerology claims that numbers have certain properties that they apply to all objects and phenomena of the world.

Even and odd numbers are used in numerology.

There are pairs of opposites in the universe, which are an important factor in its structure. The main properties that numerology ascribes to odd (1, 3, 5, 7, 9) and even (2, 4, 6, 8) numbers as pairs of opposites are as follows:

1 - active, purposeful, imperious, callous, leading, initiative;

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, enterprising, nervous, insecure;

6 - simple, calm, homely, arranged; mother's love;

7 - departure 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 the energy of "1", the brilliance and luck of "3", the adventurous mobility and versatility of "5", the wisdom of "7" and the perfection of "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, masculine and good was associated with odd numbers; many, left, feminine and evil - with even.

The masculine properties of odd numbers stem from the fact that they are stronger than even numbers. If an even number is split in half, then, apart from emptiness, nothing will remain in the middle. An odd number is not easy to split because there is a dot in the middle. If you add together an even and an odd number, then the odd one wins, since the result will always be odd. That is why odd numbers have masculine properties, imperious and sharp, and even numbers - feminine, passive and perceiving.

Odd numbers odd number: there are five of them. Even numbers an even number - four.
Odd numbers are solar, electric, 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 are opposed to 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. But when hostile numbers are in the same combination, they make each other on their guard and encourage active action; thus, these two people work a lot more. In this case, hostile numbers turn out to be really friends, and friends are real enemies, hindering progress. Neutral numbers remain inactive. They do not give support, do not cause or suppress activity.

Conclusion.

In the course of work, I found out that it was not for nothing that Pythagoras said “Number is everything.” Odd numbers, especially 3 and 7, were a symbol of fullness, happiness. Fairy tales often have three characters. And it is no coincidence that there are seven colors in the rainbow, there are seven wonders of the world in the world, there are 7 days in a week. The Bible mentions seven lamps, seven angels, seven years of abundance and seven years of famine.

Since ancient times, there have been customs to give an even or odd number of flowers, although in different countries in different ways.

And I also found out that there is a whole science that deals with numbers. This is numerology. Numerical coincidences are found all the time - in phone and car numbers, in addresses and floor numbers, in dates of birth. These are not random coincidences, but a well-defined dependence, the so-called magic of numbers. Numbers not only allow us to measure quantities, but also indicate properties and qualitative characteristics, draw our attention to various phenomena and can tell a lot. Numbers in a magical, inevitable way affect our lives, on a variety of events, and it is impossible to deny that the magic of numbers exists. You just need to find the key to their secret code.

I realized that by studying numbers and their role, you can better understand the history of your people from fairy tales. Knowing the date of birth, you can determine the character of a person. Therefore, I really enjoyed working on this essay.

Literature.

Pythagoras of Samos (570-490 BC) - ancient Greek philosopher, mathematician and mystic, creator of religious - philosophical school Pythagoreans.

For numerologists, there are only nine numbers that are involved in all calculations of the material world. All numbers above 9 just repeat them. By a simple addition method, they are reduced to single integers. For example, the number 10 is not an integer, but just a 1 followed by zero.

Zero is not a number and has no numerological value. In the Western occult tradition, zero is considered a symbol of eternity. It is surprising to learn that zero first appeared in the Western world only a few centuries ago. Its introduction greatly helped the development of mathematics, science, and modern technology. In the east, where it has been known since the dawn of civilization, zero is known as shunya or emptiness, which is the basis of Buddhism. When zero is one, it has no value because it is abstract and numbers are concrete. When zero is combined with a number, it gives birth to arithmetic progressions and series of doubles, triples, and plurals such as 10, 100, 1000. If you don't know anything about zero, you can't work with numbers above 9 (that is, leaving beyond the material world). If you know about him, his mystical nature will take you to eternity and damage your
material progress. Zero is considered unlucky. When a zero appears in a date of birth it brings bad luck. Even the tenth month of the year (October), being the 10th, brings bad luck, albeit to a small extent. The appearance of zero in the year of birth also brings bad luck - but to an even lesser extent. The combination of zero with another number reduces the influence of that number. People who have a zero in their date of birth, in general, have to fight more in their lives than those who do not have a zero. The presence of more than one zero in the date of birth - for example, October (tenth month) 10; 1950 - forces you to work very hard in life. Zero contains all the numbers from 1 to 9, and when zero connects with these numbers, a whole special series of numbers develops. For example, when zero is combined with the number 1, a series of numbers from 11 to 19 is formed. The introduction of zero to advance mathematics, general science, and modern technology led humanity into the computer age, but zero itself does not "exist."

Even and odd numbers
Numbers divided into two main groups
ODD: 1, 3, 5, 7, 9 and EVEN: 2, 4, 6, 8
Odd numbers odd number; there are five of them. Even numbers are an even number, four.
Odd numbers are solar, masculine, electrical, acidic and dynamic. They are terms (they are added to something).
Even numbers are lunar, feminine, 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
Cool. 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.
In general, 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 (see later chapters). When two friendly numbers touch, their cooperation is not very productive. Like friends, they relax - and nothing happens. But when hostile numbers are in the same combination, they make each other on their guard and encourage active action; thus, these two people work a lot more. In this case, hostile numbers turn out to be really friends, and friends are real enemies, hindering progress.
Neutral numbers remain inactive. They do not give support, do not cause or suppress activity.

Universal friend
NUMBER 6 is unique in that it is common to both odd and even numbers. It can be the result of a combination of three (3 is an odd number) even numbers, or two (2 is an even number) odd numbers. In the combination 2+2+2=6 the even number 2 is repeated three times; it is an odd number
repetitions. In the combination 3+3=6, the odd number 3 is repeated twice, here an even number of repetitions.
Being common to both groups, the number 6 is thus known as the universal friend.
9 single numbers.
There are nine single numbers. The ratio of numbers to planets is the key of numerology. In the Hindu system these relations are the same as in the Western system, with the following two exceptions. The number 4 in the Hindu system corresponds to Rahu (the north pole of the Moon), while in the Western system this number refers to the Moon and Uranus. The number 7 in the Hindu system refers to Ketu (the south pole of the moon), while in the Western system it refers to the Moon and Neptune. The nature and behavior of numbers follows from the ruling planets:
planet quality number
Sun I royalty (king), kindness,
splendor, discipline, authoritarianism, strength, originality
Moon 2 royalty (queen), attractiveness,
volatility, delicacy
Jupiter 3 spirituality, tendency to give advice,
friendliness, focus, discipline
Rahu 4 rebelliousness, impulsiveness, irascibility,
secrecy
Mercury 5 splendor, love of entertainment,
cunning, intelligence, sensitivity
Venus 6 romance, slowness, sensuality,
ability to speak, diplomacy, ingenuity
Ketu 7 mysticism, daydreaming, intuition,
ingenuity
Saturn 8 wisdom, malevolence, diligence,
obsequiousness, suffering, militancy
Mars 9 strength, rudeness, militancy, simplicity,
self-improvement, suspiciousness, struggle, alienation, discrimination between good and bad
Each person is influenced by three numbers: soul, name and fate. The influence of these numbers is different from the influence of the nine planets in the astrological houses. The influence of the Sun itself, for example, varies depending on the house and the zodiacal sign in which it is located in natal chart birth. With a change in the sign of the Sun, human behavior also changes.
In numerology, all people with soul number 1 have the qualities of this number (1) - in accordance with the month in which they were born. Differences in the month, Moon sign, Sun sign and ascension only change the direction of their behavior.
All people who have 1 ("ones") as their number have the same auspicious days, dates and years of life; they also share the same colors, stones, diets and mantras. In astrology, on the contrary, the strength of the planets and, accordingly, their management of numbers varies depending on which house they are in. For example, the rising of the Sun in the position of Aries in the eighth or twelfth house becomes fruitless because these positions are located in inauspicious houses. A similar position of the Sun in Aries becomes simply wonderful
noah in the tenth house. Similarly, the ascension of Saturn is unfavorable in the third, sixth, ninth or eleventh house and so on. Astrology is a more exact science than numerology. Such specific details help the astrologer in understanding the status of the individual. Numerology is a more general teaching and only considers the behavioral aspect of the human personality. It has developed its own language, which refers to the discussion of the personal qualities of a person. Numerology is also easier to learn than astrology. It is easy enough to memorize certain things without going into too much detail, such as the movements of the planets. Numerology is a science accessible to everyone.

All integers from the point of view of divisibility by 2 are divided into two sets: set of even numbers And set of odd numbers.

Even numbers are evenly divisible by 2, and odd when divided by 2, the remainder is 1. 0 – the number is even.

When solving problems that use the parity property, it is important to remember and apply the following rules:

• Sum and Difference two odd numbers is even number
• Sum and Difference two even numbers is even number.
• The sum and difference of two numbers, of which one even, A other odd, is odd number.
• Work two odd numbers is odd number.
• The product of two numbers, of which one even, is even number.

Let's look at a few examples.

Task 1.

Is it possible to exchange 25 rubles with ten banknotes of denominations of 1, 3 and 5 rubles?

Solution.

It is forbidden. And not because such bills do not exist. The sum of an even number of odd terms cannot be an odd number.

Answer: You can't.

Task 2.

The set contained 23 weights weighing 1 kg, 2 kg, 3 kg, ... 23 kg. Is it possible to decompose them into two parts equal in mass if a weight of 21 kg is lost?

Solution.

The mass of all weights S = (1 + 23) + (2 + 22) + ... + (11 + 13) + 12 is an even number.

Therefore, (S - 21) cannot be decomposed into two parts equal in weight, since this number is odd.

Answer. 23 weights with a given mass cannot be decomposed into two equal parts.

Task 3.

The grasshopper jumps in a straight line in different directions: the first jump is 1 cm, the second is 2 cm, the third is 3 cm, and so on. Can he, after the twenty-fifth jump, return to the point from which he started?

Solution.

Let the grasshopper jump along a number line in different directions and start from a point with coordinate 0. After 25 jumps, he will end up at a point with an odd coordinate (among numbers from 1 to 25 – odd – odd number). Since 0 is an even number, it cannot return to its original position.

Answer. After 25 jumps, the grasshopper cannot return to the point from which it started.

Task 4.

An ancient manuscript contains a description of a city located on 8 islands. The islands are connected to each other and to the mainland by bridges. 5 bridges go to the mainland; 4 islands have 4 bridges each, 3 islands have 3 bridges each, and one island can only be reached by one bridge. Could there be such an arrangement of bridges?

Solution.

Find the number of ends for all bridges:

5 + 4 4 + 3 3 + 1 = 31.

31 – is an odd number.

Since the number of ends of all bridges must be even, there cannot be such an arrangement of bridges.

Answer. Can not.

Task 5.

There are 6 glasses on the table. Of these, 5 glasses stand correctly, and one – turned upside down. It is allowed to turn over any 2 glasses in one move. Is it possible to place all glasses correctly in a finite number of moves?

Solution.

To solve this problem, let's try to formulate the condition in the language of numbers. For this, the event "the glass is standing correctly" will be numbered 1, and "the glass is not standing correctly" – 0. Then instead of drawing with glasses there will be a sequence of five ones and one zero. The sum of all numbers in the sequence is equal to the odd number 5. When the glass is turned over in our sequence, 0 will change to 1 and vice versa - 1 to 0. Our goal is to get a row of only 1. There should be 6 of them and the sum should also be equal to 6. This number even.

But what happens to the sum when you flip 2 glasses at the same time? Either two 1s are replaced by 0s, or two 0s by ones, or one 1 by 0 and one 0 by 1. What happens to the sum? In the first and second cases, it changes by 2, and in the third, it does not change at all. And this means that it will never become even and can never become equal to 6, as, by the way, neither 2 nor 4.

Answer. Impossible.

Task 6.

Petya bought a common notebook with a volume of 96 sheets and numbered all its pages in order with numbers from 1 to 192. Vasya tore out 25 sheets from this notebook and added up all 50 numbers that are written on them. Could he get the number 2006?

Solution.

Let's pay attention to the sum of page numbers on one sheet. It is odd, because one page corresponds to an odd number, and the second page of the sheet has an even number. But there are 25 sheets. Then the sum of all numbers of torn pages is odd. And what did Vasya get? Therefore, he is wrong!

Answer. Couldn't.

Task 7.

Each of the 10 numbers is written on a card. 2 such sets were made. We received 20 cards, each of which has the number 0 or 1 or 2 ... or 9 and cards with the same numbers by 2. Prove that it is impossible to arrange these cards in one row so that there are exactly k cards between identical cards with number k. (k = 0, 1, 2, …, 9).

Solution.

Let's assume that it was possible to decompose the cards in the indicated way. Then it is easy to number them in order by numbers from 1 to 20. Suppose that each first card with the number k in the row has the number a k and the last one with the same number k has the number b k . Then b k – and k = k + 1. Then

∑(b k – a k) = ∑b k – ∑a k \u003d (b 0 - a 0) + (b 1 - a 1) + (b 2 – a 2) + (b 3 – a 3) + ... + (b 9 – a 9) = 1 + 2 + 3 + 4 + ... + 10 = 55.

But ∑b k + ∑a k = 1 + 2 + 3 + ... + 20 = 210. (The sum of all card numbers.).

We got ∑b k – ∑a k = 55 and ∑b k + ∑a k = 210. Adding these equalities, we get 2∑b k = 265, which is impossible. (In all cases, the sign ∑ means summation over k from 0 to 9.) The number on the right is even, and on the left, it is odd. This contradiction proves that our assumption that it is possible to arrange the cards in this way is erroneous.

Answer. The assertion has been proven.

If you have mastered the material in this article well, then solving the following problems should not cause any particular difficulties for you. In case of difficulty, try to find among the solved problems of related content.

1. 8 raspberry bushes grow along the fence. The number of berries on neighboring bushes differs by one. Can all bushes together have 225 berries?
2. There are 1,001 cities in the Kingdom. The king ordered that roads be laid between the cities so that 7 roads came out of each city. Will the subjects be able to cope with the order of the king?

I wish you success!

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An integer is said to be even if it is divisible by 2; otherwise it is called odd. So the even numbers are

and odd numbers

From the divisibility of even numbers by two, it follows that every even number can be written as , where the symbol denotes an arbitrary integer. When a symbol (like a letter in our case) can represent any element of some particular set of objects (the set of integers in our case), we say that the range of this symbol is the specified set of objects. In accordance with this, in the case under consideration, we say that every even number can be written in the form , where the range of the symbol coincides with the set of integers. For example, the even numbers 18, 34, 12, and -62 have the form , where they are 9, 17, 6, and -31, respectively. There is no particular reason to use the letter here. Instead of saying that even numbers are integers of the form one could equally say that even numbers are of the form or or

When two even numbers are added together, the result is also an even number. This circumstance is illustrated by the following examples:

However, a set of examples is not enough to prove the general assertion that the set of even numbers is closed under addition. To give such a proof, let's denote one even number by , and the other by . Adding these numbers, we can write

The sum is written as . This shows that it is divisible by 2. It would not be enough to write

since the last expression is the sum of an even number and the same number. In other words, we would prove that twice an even number is again an even number (in fact divisible even by 4), while we need to prove that the sum of any two even numbers is an even number. Therefore, we have used the notation for one even number and for another even number in order to indicate that these numbers can be different.

What notation can be used to write any odd number? Note that subtracting 1 from an odd number results in an even number. Therefore, it can be argued that any odd number is written in the form A record of this kind is not unique. Similarly, we might notice that adding 1 to an odd number results in an even number, and we might conclude from this that any odd number can be written as

Similarly, we can say that any odd number is written as or or, etc.

Can it be argued that every odd number is written as Substituting into this formula instead of integers

we get the following set of numbers:

Each of these numbers is odd, but they do not exhaust all odd numbers. For example, the odd number 5 cannot be written this way. Thus, it is not true that every odd number has the form , although every integer of the form is odd. Similarly, it is not true that every even number is written as where the range of the symbol k is the set of all integers. For example, 6 is not equal to whichever integer you take as A. However, every integer of the form is even.

The relation between these statements is the same as between the statements "all cats are animals" and "all animals are cats". It is clear that the first of them is true, but the second is not. This relationship will be discussed further in the analysis of statements that include the phrases "then", "only then" and "then and only then" (see § 3 ch. II).

Exercises

Which of the following statements are true and which are false? (It is assumed that the range of characters is the collection of all integers.)

1. Every odd number can be represented as

2. Every integer of the form a) (see exercise 1) is odd; the same holds for numbers of the form b), c), d), e) and f).

3. Every even number can be represented as

4. Every integer of the form a) (see exercise 3) is even; the same holds for numbers of the form b), c), d) and e).

· Even numbers are those that are divisible by 2 without a remainder (for example, 2, 4, 6, etc.). Each such number can be written as 2K by choosing a suitable integer K (for example, 4 = 2 x 2, 6 = 2 x 3, etc.).

· Odd numbers are those that, when divided by 2, give a remainder of 1 (for example, 1, 3, 5, etc.). Each such number can be written as 2K + 1 by choosing a suitable integer K (for example, 3 = 2 x 1 + 1, 5 = 2 x 2 + 1, etc.).

• Addition and subtraction:

• • Hexact ± H ethnoe = H ethnoe
  • Hexact ± H even = H even
  • Heven ± H ethnoe = H even
  • Heven ± H even = H ethnoe
• Multiplication:
• • Hblack × H ethnoe = H ethnoe
  • Hblack × H even = H ethnoe
  • Heven × H even = H even
• Division:
• • Hethnoe / H even - it is impossible to unambiguously judge the parity of the result (if the result integer, it can be either even or odd)
  • Hethnoe / H even --- if result integer, then it H ethnoe
  • Heven / H parity - the result cannot be an integer, and therefore have parity attributes
  • Heven / H even --- if result integer, then it H even

The sum of any number of even numbers is even.

The sum of an odd number of odd numbers is odd.

The sum of an even number of odd numbers is even.

The difference of two numbers is the same parity as their sum.
(ex. 2+3=5 and 2-3=-1 are both odd)

Algebraic (with + or - signs) sum of integers It has the same parity as their sum.
(e.g. 2-7+(-4)-(-3)=-6 and 2+7+(-4)+(-3)=2 are both even)

The idea of ​​parity has many different applications. The simplest of them:

1. If objects of two types alternate in some closed chain, then there are an even number of them (and of each type equally).

2. If objects of two types alternate in some chain, and the beginning and end of the chain different types, then it has an even number of objects, if the beginning and end of the same type, then an odd number. (an even number of objects corresponds to odd number of transitions between them and vice versa !!! )

2". If the object alternates between two possible states, and the initial and final states different, then the periods of the object's stay in one state or another - even number, if the initial and final states are the same - then odd. (reformulation of paragraph 2)

3. Conversely: by the evenness of the length of an alternating chain, you can find out whether its beginning and end are of one or different types.

3". Conversely: by the number of periods of the object's stay in one of the two possible alternating states, one can find out whether the initial state coincides with the final one. (reformulation of paragraph 3)

4. If objects can be divided into pairs, then their number is even.

5. If for some reason it was possible to divide an odd number of objects into pairs, then one of them will be a pair to itself, and there may be more than one such object (but there are always an odd number of them).

(!) All these considerations can be inserted into the text of the solution of the problem at the Olympiad, as obvious statements.

Examples:

Task 1. On the plane there are 9 gears connected in a chain (the first with the second, the second with the third ... the 9th with the first). Can they rotate at the same time?

Solution: No, they can't. If they could rotate, then two types of gears would alternate in a closed chain: rotating clockwise and counterclockwise (it does not matter for solving the problem, in which one direction of rotation of the first gear ! ) Then there should be an even number of gears, and there are 9 of them?! h.i.d. (sign "?!" means getting a contradiction)

Task 2. Numbers from 1 to 10 are written in a row. Is it possible to place + and - signs between them to get an expression equal to zero?
Solution: No you can not. Parity of the resulting expression Always will match parity amounts 1+2+...+10=55, i.e. sum will always be odd . Is 0 an even number? h.t.d.