34 is an even number. Even and odd numbers. And again about the amount and product

Evenness (oddness) considerations are often used in solving mathematical problems (both elementary and very "advanced"). This article discusses approaches to solving such problems.

We will start with the simplest examples, and in the final part we will consider several "olympiad" problems, in the solution of which we will be helped by considerations of parity.

Even and odd numbers. Initial information

In this article, we will consider mainly natural or whole numbers. Let me remind you that a number is called even if it is divisible entirely by 2. In other words, any even number n can be represented as n = 2k, where k is an integer, and any odd number can be represented as n = 2k + 1 (or n = 2k - 1). Zero, of course, will be considered an even number.

Example 1... Represent the numbers 34 and 171 as 2k or 2k + 1, where k is an integer.

34 = 2 17 (34 is an even number); 171 = 2 85 + 1 (171 is an odd number).

Exercise 1... The numbers 68, 133, -2246 and -8977 are represented as 2k or 2k + 1, where k is an integer.

Assignment 2... Imagine the number 18 as: a) the sum of two even numbers, b) the sum of two odd numbers. Can you get 18 when adding odd and even numbers?

Assignment 3... Imagine the number 24 as: a) the product of two even numbers, b) the product of an even and an odd number. Can you get 24 when multiplying two odd numbers?

Sum, product, quotient of even (odd) numbers

Statement 1... The sum of two even numbers is an even number.

Proof. Let the numbers m and n be even. Let us prove that the number r = m + n is also even. m = 2k, n = 2p, where k and p are integers. Then r = m + n = 2k + 2p = 2 (k + p) = 2s. If the numbers k and p are integers, then their sum s is also an integer. We have proved that the number r can be represented as a product of two and an integer. The proof is complete.

Statement 2... The sum of two odd numbers is an even number. Prove yourself.

Statement 3... The sum of even and odd numbers is an odd number. Prove yourself.

Statement 4... The product of two odd numbers is an odd number.

Proof. Let the numbers m and n be odd. Let us prove that the number r = m n is also odd.
m = 2k + 1, n = 2p + 1, where k and p are integers.
Then r = m n = (2k + 1) (2p + 1) = 4kp + 2k + 2p + 1 = 2 (2kp + k + p) + 1 = 2s + 1.

If the numbers k and p are integers, then the number s = 2kp + k + p is also an integer.
We have proved that the number r can be represented in the form r = 2s + 1, therefore, it is odd. Ch. Ect.

Statement 5... The product of two even numbers is an even number. Prove yourself.

Statement 6... The product of an even and an odd number is an even number. Prove yourself.

What if we divide an even number by an even number (not equal to zero)? What do we get: even or odd? Naturally, there is no definite answer. For example, dividing 12 by 4 gives an odd result, and dividing 32 by 4 gives an even result.

If you are already bored, skip to the 2nd part of the article. Then you can always come back. If you are not too tired of all these theoretical constructions, let's continue.

And why, in fact, we are considering only two numbers. Let's think wider!

Statement 7... The sum of any number of even numbers is even.

Proof. Let the numbers M 1, M 2, ..., MN be even, then they can be represented as 2K 1, 2K 2, ..., 2K N, where K 1, K 2, ..., KN are integers ...

Then: M 1 + M 2 + ... + M N = 2K 1 + 2K 2 + ... + 2K N = 2 (K 1 + K 2 + ... + K N) = 2S, where S is an integer. Parity is proven.

Statement 8... The sum of an even number of odd numbers is even. The sum of an odd number of odd numbers is odd. Prove yourself.

Statement 9... The product can be odd only if all factors are odd. Prove yourself.

So, the sum 2 + 4 + 6 + ... + 1022 + 1024 is even, since all the terms are even. The sum 1 + 3 + 5 + 7 + 9 is odd since it contains 5 odd terms. The product 2 * 3 * 4 * ... * 1001 * 1002 is even, if only for the reason that the first factor is even.

Assignment 4... The following expressions will be even or odd: a) 2 + 12 + 22 + ... + 1002 + 1012 + 1022, b) 1 + 11 + 111 + ... + 111111 + 1111111, c) 3 * 13 * 23 *. .. * 10003 * 10013 * 10023, d) 2 * 3 * 4 * ... * 12357891?

Assignment 5... Prove that the product of all primes up to 1,000,000 is even. Prove that the product of any number of primes, each of which is greater than 100, is odd. Let me remind you that natural number is called simple if it is divisible only by itself and by 1.

And again about the amount and product

Example 2... Young mathematician Petya added the sum of two integers and their product. He claims that he got the number 56792. Is this possible if it is known that at least one of the original numbers is odd?

Solution. Let's denote the initial numbers A and B. Obviously, 4 options are possible:

A and B are even numbers (but this case is not considered in the problem),
A and B are odd numbers,
A is even and B is odd,
A is odd, B is even.

In principle, the last two cases could be painlessly combined, but for us this is not essential now. In the previous paragraph, we found out everything about the parity of the sum and the product. Now let's put together a table. In the first two columns we indicate the parity of the numbers A and B, in the 3rd column - the parity of the sum, in the 4th the parity of the product, in the 5th - the parity of the final number.

A	B	A + B	AB	(A + B) + AB
H	H	H	H	H
H	H	H	H	H
H	H	H	H	H
H	H	H	H	H

In all cases (except the first one) we obtain odd result!

By the way, our young friend Petya claims that he received an even number. We have proven that this is impossible. Petya was wrong.

Assignment 6... The young mathematician Masha multiplied the product of two integers by their sum. She claims that the number is 89999719. Is Masha right?

Assignment 7... Young mathematician Petya claims that when adding two integers he got 927, and when multiplying - 6321. Is this possible? Explain your answer.

I am aware that the first part of the article may seem rather tiresome and monotonous to the reader. Unfortunately, these "boring" basic concepts cannot be dispensed with. I promise it will be much more interesting.

Odd number is an integer that does not share with no remainder:…, −3, −1, 1, 3, 5, 7, 9, ...

If m is even, then it can be represented as $m = 2 k$ , and if it is odd, then in the form $m = 2 k + 1$ , where $k \ in \ mathbb Z$ .

History and culture

The concept of evenness of numbers has been known since ancient times and it was often given a mystical meaning. In Chinese cosmology and natural philosophy, even numbers correspond to the concept of "yin", and odd numbers correspond to "yang".

IN different countries there are traditions associated with 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 funeral of the deceased. 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 lady a bouquet of 12, 14, 16, etc. flowers or sections of a bush flower that have many buds, in which they, in principle, are not counted. Moreover, this applies to a large number of flowers (cuts) given in other cases.

Practice

In higher education institutions with complex schedules educational process even and odd weeks apply. Within these weeks, the timetable for training sessions and, in some cases, the start and end times are different. This practice is used to evenly distribute the load in classrooms, educational buildings and for the rhythm of classes in disciplines with a low classroom load (1 time in 2 weeks)

In train schedules, even and odd train numbers are used, depending on the direction of movement (forward or backward). Accordingly, evenness / oddness denotes 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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An excerpt characterizing Even and Odd numbers

- So, so, - said Prince Andrey, referring to Alpatych, - tell everything as I told you. And, not 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 followed them. On August 10, the regiment, commanded by Prince Andrey, passed along the high road, past the avenue leading to Lysye Gory. The heat and drought lasted for over three weeks. Curly clouds walked across the sky every day, occasionally blocking the sun; but in the evening it cleared again, and the sun set in a brownish-red haze. Only the strong dew at night refreshed the earth. The bread remaining at the root burned and poured out. The swamps are dry. The cattle roared with hunger, finding no food on the meadows burned by the sun. Only at night and in the forests there was still dew, 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 had been pounded by more than a quarter of an arshin. As soon as dawn dawned, movement began. Carts, artillery silently walked along the hub, and the infantry was ankle-deep 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 into the eyes, hair, ears, nostrils and, most importantly, into 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 in the sun, not covered by clouds, one could see with a simple eye. The sun appeared to be a large crimson ball. There was no wind and people were suffocating in this still atmosphere. People walked with handkerchiefs tied around their noses and mouths. Coming to the village, everything rushed to the wells. They fought for water and drank it to the mud.
Prince Andrey commanded the regiment, and the structure of the regiment, the well-being of its people, the need to receive and issue orders occupied him. The fire of Smolensk and its abandonment were an era for Prince Andrei. A new feeling of bitterness against the enemy made him forget his grief. He was all devoted to the affairs of his regiment, he was caring about his people and officers and kindness to 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 regiments, with Timokhin, etc., with people completely new 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, from the staff, he immediately bristled again; became spiteful, 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 seemed to Prince Andrei in a dark, gloomy light - especially after they left Smolensk (which, in his opinion, could and should have been defended) on August 6, and after the sick father had to flee to Moscow and throw the Bald Hills so beloved, built and inhabited by them, to plunder; but in spite of this, thanks to the regiment, Prince Andrey could think about another subject, completely independent of general questions - about his regiment. On August 10, the column, in which his regiment was, drew level with the Bald Mountains. Prince Andrey two days ago received news that his father, son and sister had left for Moscow. Although Prince Andrey had nothing to do in Bald Hills, he, with his usual desire to squander his grief, decided that he should stop by in Lysy Gory.
He ordered to saddle his horse and from the crossing rode on horseback to his father's village, in which he was born and spent his childhood. Driving past the pond, on which dozens of women were always chatting, beating with rollers and rinsing their linen, Prince Andrey noticed that there was no one on the pond, and a torn raft, half flooded with water, was floating sideways in the middle of the pond. Prince Andrew drove up to the gatehouse. There was no one at the stone gate of the entrance, and the door was unlocked. The garden paths were already overgrown, and the calves and horses were walking in the English park. Prince Andrew drove up to the greenhouse; the windows were broken, and some of the trees in tubs were knocked down, some were withered. He called out to Taras the gardener. Nobody responded. Turning around the greenhouse to the exhibition, he saw that the carved board fence was all broken and the plum fruit had been torn off with branches. An old peasant (Prince Andrey had seen him at the gate as a child) was sitting and weaving bast shoes on a green bench.
He was deaf and did not hear the entrance of Prince Andrew. He was sitting on a bench, on which the old prince liked to sit, and near him there was a little mark on the twigs of a broken and dried magnolia.
Prince Andrew drove up to the house. Several lindens in the old garden were cut down, and one horse with a skewbald colt walked in front of the house between the rose trees. The house was boarded up with shutters. One window at the bottom was open. The yard boy, seeing Prince Andrey, ran into the house.
Alpatych, having sent his family, remained alone in the Bald Mountains; he sat at home and read the Life. Having learned about the arrival of Prince Andrey, he, with glasses on his nose, buttoning himself up, left the house, hurriedly went up to the Prince and, without saying anything, wept, kissing Prince Andrey on the knee.

What do even and odd numbers mean in spiritual numerology. This is a very important topic in the study! How do even numbers differ from odd numbers in their ESSENCE?

Even numbers

It is common knowledge that even numbers are those that are divisible by two. That is, the numbers 2, 4, 6, 8, 10, 12, 14, 16, 18 and so on.

And what do the even numbers mean relative? What is the numerological essence of dividing by two? And the point is that all numbers that are divisible by two carry some properties of two.

It has several meanings. Firstly, this is the most "humane" number in numerology. That is, the number 2 reflects in itself the whole gamut of human weaknesses, shortcomings and advantages - more precisely, what society is considered to be advantages and disadvantages, "correctness" and "incorrectness".

And since these labels of "correctness" and "incorrectness" reflect our limited views on the world, then two can be considered the most limited, the most "dumb" number in numerology. Hence, it is clear that even numbers are much more "die-hard" 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 simply different and reflect different 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 there is another meaning of two: standard logical thinking. And all even numbers in spiritual numerology, in one way or another, obey certain logical rules for perceiving reality.

An elementary example: if a stone is thrown up, it, having gained a certain height, then rushes to the ground. Even numbers “think” this way. And odd numbers will easily assume that the stone will fly into space; or it will not fly, but will get stuck somewhere in the air ... for a long time, for centuries. Or just dissolve! The more illogical a hypothesis is, the closer it is to odd numbers.

Odd numbers

Odd numbers are 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 obey not material, but spiritual logic.

Which, by the way, gives food for thought: why is the number of flowers in a bouquet for a living person odd, and for a dead person - even ... Is it because material logic (logic in the framework of "yes-no") is dead relative to the human soul?

Apparent coincidences of material logic and spiritual logic 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 every step of the way. We admit it, dream about it, decorate our life and someone else's life with it.

But what do we really know about love? About that all-pervading Love that permeates all spheres of the Universe. Can we agree and accept that there is as much cold as warmth, as much hatred as kindness in her ?! Are we able to realize that these paradoxes constitute the highest, creative essence of Love ?!

Paradox is one of the key properties of odd numbers. IN interpretation of odd numbers one must understand: not always what seems to a person is 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, maturity of the mind is characterized by the ability to perceive paradoxes. Therefore, it takes a little more brainpower 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 (other than 10), solid and consistent. Events and people associated with even numbers are more stable and explicable. They are quite accessible for external changes, but only for external ones! Internal changes are the realm of odd numbers ...

Odd numbers are flighty, freedom-loving, unstable, unpredictable. They always bring surprises. It seems like you know the meaning of some odd number, but it, this number, suddenly begins to behave in such a way that it makes you reconsider almost your whole life ...

Note!

My book, Spiritual Numerology. The language of numbers. " Today it is the most complete and popular of all existing esoteric textbooks on the meaning of numbers. More about this,and also to order the book, follow the link below: « «

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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, domineering, callous, leading, proactive 2 - passive, receptive, weak, sympathetic, subordinate 3 - bright, cheerful, artistic, successful, easily achieving success 4 - hardworking, boring, lack of initiative, unhappy; hard work and frequent defeat 5 - agile, enterprising, nervous, insecure, sexy 6 - simple, calm, homely, arranged; maternal love 7 - leaving the world; mysticism, secrets 8 - worldly life; material success or failure 9 - intellectual and spiritual perfection

Odd numbers have much more striking properties. Next to the energy of "1", the brilliance and luck of the "3", the adventurous mobility and versatility of the "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, masculine - feminine, good - evil. One, right, masculine and good was associated with odd numbers; many, left, feminine and evil - with even ones. Odd numbers have a certain productive mean, while any even number has a perceiving hole, as it were, a lacuna within itself. The masculine properties of phallic odd numbers stem from the fact that they are stronger than even ones. If an even number is split in half, then, apart from the emptiness, there will be nothing left in the middle. It is not easy to split an odd number because a dot remains in the middle. If you put 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 qualities, domineering and harsh, and even ones - feminine, passive and perceiving. Odd numbers are an odd number: there are five of them. Even numbers are even number - four. Odd numbers are solar, electric, acidic and dynamic. They are addends; they are added to anything. 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 (odd two or even two) 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 govern them (details in the section "Number Compatibility"). When two friendly numbers touch, their collaboration is not very productive. Like friends, they relax - and nothing happens. But when there are hostile numbers in one combination, they make each other be on the alert and induce active action; so these two people work a lot harder. In this case, hostile numbers turn out to be in fact friends, and friends are real enemies, hindering progress. Neutral numbers remain inactive. They do not provide support, induce or suppress activity.