Alexander Domogarov announced that he was leaving the theater. Divisibility of integers and remainders - copy from Antoshka

If two numbers A And b when dividing by a number m give identical remainders, then we say that a is comparable to b modulo m. Write it down like this a ≡ b (mod m)

If a>b, then the greatest common divisor a And b equal to greatest common divisor a–b And b.

Let's consider these properties when solving problems:

1. How many natural numbers are there less than 1000 that are not divisible by either 5 or 7?

Solution: From 999 numbers less than 1000, we cross out numbers that are multiples of 5: there are 199 of them (999/5 = 199). Next, we cross out numbers that are multiples of 7: there are 142 of them (999/7 = 142). But among the numbers that are multiples of 7, there are 28 (999/35 = 28) numbers that are also multiples of 5; they will be crossed out twice. In total, we must cross out 199 + 142 – 28 = 313 numbers.

That leaves 999 – 313 = 686. Answer: 686 numbers.

2. Find the remainder of 2009⋅2010⋅2011+2012 divided by 2 by 7.

The solution of the problem

Considering that 2009⋮7, the remainder will be equal to 2012 2 ≡ 3 2 ≡ 2(mod7)

3. It is known that the remainder of the number aa divided by 19 is equal to 7, and the number b by 19 is equal to 11. Find the remainder of the number ab(a+b)(a−b) when divided by 19.

The solution of the problem

Note that ab(a+b)(a−b)≡ 7⋅11⋅18⋅(−1) ≡ 7⋅(−8)⋅(−1)⋅(−4) =−224 = −228+4 ≡ 4(mod19)

4. Prove that the sum of the squares of three integers cannot leave a remainder of 7 when divided by 8.

Solution

Any integer when divided by 8 has a remainder of one of the following eight numbers 0, 1, 2, 3, 4, 5, 6, 7, so the square of an integer has a remainder of one of the three numbers 0, 1, 4 when divided by 8. In order for the sum of the squares of three numbers to have a remainder of 7 when divided by 8, it is necessary that one of two cases be true: either one of the squares or all three have odd remainders when divided by 8.

In the first case, the odd remainder is 1, and the sum of two even remainders is 0, 2, 4, that is, the sum of all remainders is 1, 3, 5. The remainder 7 in this case cannot be obtained. In the second case, three odd remainders are three 1s, and the remainder of the entire sum is 3. So, 7 cannot be a remainder when dividing by 8 the sum of the squares of three integers.

5. Are there natural nn such that n 2 +n+1 is divisible by 2014?

The solution of the problem

Note that n 2 + n = n(n+1) is divisible by 2, since it is the product of two consecutive numbers, which means n 2 + n+1 is always odd (this could also be noticed using Fermat’s little theorem: n 2 + n + 1 ≡ n + n+1 = 2n + 1 ≡1 (mod 2).

Since the number 2014 is even, there are no n such that the number n 2 +n+1 is divisible by 2014 (if such n existed, this would contradict the fact that n 2 +n+1 is odd).

6. C Is there a ten-digit number divisible by 11 in which each digit appears once?

Method I When writing out three-digit numbers divisible by 11, you can find three numbers among them, the recording of which involves all the numbers from 0 to 9. For example, 275, 396,418. Using them, you can create a ten-digit number divisible by 11. For example:

2753964180 = 275 107 + 396 107 + 418 10 = 11 (25 107 + 36 104 + 38 10).

II method. To find the required number, we will use the test of divisibility by 11, according to which the numbers n = a 1 a 2 a 3 ...a 10 (in this case, a i are not factors, but digits in the notation of number n) and S(n) = a 1 –a 2 +a 3 –…–a 10 are simultaneously divisible by 11.

Let A be the sum of digits included in S(n) with a “+” sign, B – the sum of digits included in S(n) with a “–” sign. The number A–B, according to the conditions of the problem, must be divisible by 11. Let B – A = 11, in addition, obviously, A + B = 1+2+3+…+9 = 45. Solving the resulting system B – A =11 , A + B = 45, we find, A = 17, B = 28. Let's select a group of five different numbers with a sum of 17. For example, 1+2+3+5+6 = 17. Let's take these numbers as numbers with odd numbers . As even-numbered digits, let’s take the remaining ones – 4, 7, 8, 9, 0.

We see that, for example, the number 1427385960 satisfies the conditions of the problem.

7. Find the smallest natural number that gives the same remainder when divided by 25 as 1234.

Solution

Let's consider the remainder when dividing the number 1234 by 25. All numbers smaller than it give other remainders, since they themselves are their own remainders. The remainder when dividing 1234 by 25 is 9, since 1234=49⋅25+9, this will be the answer.

8. Having received a bad mark in geography, Vasya decided to tear the geographical map to shreds. He tears every scrap that comes into his hands into four pieces. Can he ever get exactly 2012 pieces? 2013 pieces? 2014 pieces? 2015 pieces?

The solution of the problem

Note that each time Vasya increases the number of pieces by 3, since he turns one piece into four. Therefore, he will receive numbers of the form 1+3N, where N is the number of pieces that he tore into pieces. The number 2014 has this form, so it will have 2014 pieces, but others cannot be represented in this form (their remainders when divided by 3 are 0 or 2).

9. Find the smallest natural number that gives the following remainders: 1 - when divided by 2, 2 - when divided by 3, 3 - when divided by 4, 4 - when divided by 5, 5 - when divided by 6.

The solution of the problem

Consider the desired number increased by one. It is divisible by 2,3,4,5,6, because it gives remainders that are less by one than the divisors themselves. We need to find the minimum such number, therefore, the required number is the least common multiple of the numbers 2,3,4,5,6 minus 1. The least common multiple of 2,3,4,5,6 is 2 2 ⋅3⋅5=60, because in the numbers 2,3,4,5,6 there are only 3 prime divisors, three and five are included at most in the first power, and two in the second (in the number 4). This means that the required number is 60−1 = 59.

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The numbers are written out in a row: , , …, , “+” and “−” signs are placed between them randomly and the resulting sum is found.

Could this amount be equal to:

a) −4, if ?

b) 0 if ?

c) 0 if ?

d) −3, if ?

The lengths of the sides of a rectangle are natural numbers, and its perimeter is 200. It is known that the length of one side of a rectangle is n n– also a natural number.

n>100.

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Several (not necessarily different) natural numbers are conceived. These numbers and all their possible sums (2, 3, etc.) are written on the board in non-decreasing order. If some number n written on the board is repeated several times, then one such number is left on the board n, and the remaining numbers are equal n, are erased. For example, if the numbers are 1, 3, 3, 4, then the set 1, 3, 4, 5, 6, 7, 8, 10, 11 will be written on the board.

a) Give an example of planned numbers for which the set 2, 4, 6, 8, 10 will be written on the board.

b) Is there an example of such conceived numbers for which the set 1, 3, 4, 5, 6, 8, 10, 11, 12, 13, 15, 17, 18, 19, 20, 22 would be written on the board?

c) Give all examples of conceived numbers for which the set 7, 8, 10, 15, 16, 17, 18, 23, 24, 25, 26, 31, 33, 34, 41 will be written on the board.

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The lengths of the sides of a rectangle are natural numbers, and its perimeter is 4000. It is known that the length of one side of a rectangle is n% of the length of the other side, where n- also a natural number.

a) What is the largest value the area of a rectangle can take?

b) What is the smallest value that the area of a rectangle can take?

c) Find all possible values that the area of a rectangle can take, if it is additionally known that n

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There are 8 cards. Each of the numbers 1, -2, -3, 4, -5, 7, -8, 9 is written on them one at a time. The cards are turned over and shuffled. On their blank sides, each of the numbers 1, -2, -3, 4, -5, 7, -8, 9 is written again one at a time. After this, the numbers on each card are added, and the resulting eight sums are multiplied.

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Several integers are conceived. A set of these numbers and all their possible sums (2, 3, etc.) are written on the board in non-decreasing order. For example, if the numbers are 2, 3, 5, then the set 2, 3, 5, 5, 7, 8, 10 will be written on the board.

a) The set -11, -7, -5, -4, -1, 2, 6 is written on the board. What numbers were intended?

b) For some different conceived numbers in the set written on the board, the number 0 appears exactly 4 times. What is the smallest number of numbers that could be conceived?

c) For some planned numbers, a set is written out on the board. Is it always possible to unambiguously determine the intended numbers from this set?

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Before each of the numbers 14, 15, . . ., 20 and 4, 5, . . ., 8, a plus or minus sign is arbitrarily placed, after which each of the resulting numbers of the second set is subtracted from each of the resulting numbers of the first set, and then all 35 obtained results are added. What is the smallest modulo and what is the largest sum that can be obtained in the end?

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There are 8 cards. Write each of the numbers on them one at a time:

The cards are turned over and shuffled. On their blank sides they write again one of the numbers:

−11, 12, 13, −14, −15, 17, −18, 19.

After this, the numbers on each card are added, and the resulting eight sums are multiplied.

a) Can the result be 0?

b) Could the result be 117?

c) What is the smallest non-negative integer that can result?

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The number is such that for any representation as a sum of positive terms, each of which does not exceed these terms, can be divided into two groups so that each term falls into only one group and the sum of the terms in each group does not exceed

a) Can the number be equal

b) Can the number be greater?

c) Find the maximum possible value

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An arithmetic progression (with a difference other than zero) is given, composed of natural numbers whose decimal notation does not contain the number 9.

a) Can such a progression have ten terms?

b) Prove that the number of its members is less than 100.

c) Prove that the number of terms of any such progression is not more than 72.

d) Give an example of such a progression with 72 terms

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Each of the numbers 1, −2, −3, 4, −5, 7, −8, 9 is written one at a time on 8 cards. The cards are turned over and shuffled. On their blank sides, each of the numbers 1, −2, −3, 4, −5, 7, −8, 9 is written again one at a time. After this, the numbers on each card are added, and the resulting eight sums are multiplied.

a) Can the result be 0?

b) Can the result be 1?

c) What is the smallest non-negative integer that can result in

will it work out?

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The number 7 is written on the board. Once a minute, Vasya adds one number to the board: either twice the value of one of the numbers on the board, or equal to the sum of some two numbers written on the board (thus, after one minute a second number will appear on the board number, after two - the third, etc.).

a) Could the number 2012 appear on the board at some point?

b) At some point, can the sum of all the numbers on the board equal 63?

c) In what shortest time can the number 784 appear on the board?

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Find all prime numbers b, for each of which there is an integer A that the fraction can be reduced by b.

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The natural numbers from 1 to 20 are divided into four groups, each of which contains at least two numbers. For each group, find the sum of the numbers in this group. For each pair of groups, find the modulus of the difference of the found sums and add the resulting 6 numbers.

a) Can the result be 0?

b) Can the result be 1?

c) What is the smallest possible value of the result obtained?

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Before each of the numbers 3, 4, 5, . . . 11 and 14, 15, . . . 18 randomly put a plus or minus sign, after which each of the resulting numbers of the first set is added to each of the resulting numbers of the second set, and then all 45 results obtained are added up. What is the smallest modulo sum and what is the largest sum that can be obtained in the end?

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The numbers from 10 to 21 are written once in a circle in some order. For each of the twelve pairs of adjacent numbers, their greatest common divisor is found.

a) Could it happen that all greatest common divisors are equal to 1?

b) Could it happen that all greatest common divisors are pairwise distinct?

c) What is the largest number of pairwise distinct greatest common divisors that could result?

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Each of the numbers 5, 6, . . ., 9 is multiplied by each of the numbers 12, 13, . . ., 17 and a plus or minus sign is placed in front of each arbitrary image, after which all 30 results obtained are added up. What is the smallest modulo sum and what is the largest sum that can be obtained in the end?

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Natural numbers from 1 to 21 were placed on a circle in some way (each number was placed once). Then, for each pair of adjacent numbers, we found the difference between the larger and smaller numbers.

a) Could all the resulting differences be at least 11?

b) Could all the resulting differences be at least 10?

c) In addition to the differences obtained, for each pair of numbers separated by one, we found the difference between the greater and the lesser. For what is the largest integer k you can arrange the numbers in such a way that all differences are no less k?

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a) The set −6, −2, 1, 4, 5, 7, 11 is written on the board. What numbers were intended?

b) For some different conceived numbers in the set written on the board, the number 0 appears exactly 7 times. What is the smallest number of numbers that could be conceived?

c) For some planned numbers, a set is written out on the board. Is it always possible to unambiguously determine the intended numbers from this set?

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n n n

a) Give an example of planned numbers for which the set 2, 4, 6, 8 will be written on the board.

b) Is there an example of such conceived numbers for which the set 1, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 22 would be written on the board?

c) Give all examples of conceived numbers for which the set 9, 10, 11, 19, 20, 21, 22, 30, 31, 32, 33, 41, 42, 43, 52 will be written on the board.

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Find an irreducible fraction such that

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a) What is the number of ways to write the number 1292 in the form where the numbers are integers,

b) Are there 10 different numbers such that they can be represented in the form where the numbers are integers in exactly 130 ways?

c) How many numbers N are there such that they can be represented in the form where the numbers are integers in exactly 130 ways?

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a) Give an example of planned numbers for which the set 1, 2, 3, 4, 5, 6, 7 will be written on the board.

b) Is there an example of such conceived numbers for which the set 1, 3, 4, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 19, 20, 22 would be written on the board?

c) Give all examples of conceived numbers for which the set 7, 9, 11, 14, 16, 18, 20, 21, 23, 25, 27, 30, 32, 34, 41 will be written on the board.

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Kolya multiplied a certain natural number by a neighboring natural number, and received a product equal to m. Vova multiplied some even natural number by a neighboring even natural number and obtained a product equal to n.

m And n equal 6?

m And n equal 13?

m And n?

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Among the ordinary fractions with positive denominators located between the numbers, find the one whose denominator is minimal.

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Each of the group of students went to the cinema or the theater, and it is possible that some of them could go to both the cinema and the theater. It is known that in the theater there were no more boys than the total number of students in the group who visited the theater, and in the cinema there were no more boys than in the total number of students in the group who visited the cinema.

a) Could there be 9 boys in the group, if it is additionally known that there were 20 students in total in the group?

b) What is the largest number of boys COULD be in the group, if it is additionally known that there were 20 students in total in the group?

c) What is the smallest proportion that girls could make up of the total number of students in the group without the additional condition of points a and b?

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Given a three-digit natural number (the number cannot start from zero), not a multiple of 100.

a) Can the quotient of this number and the sum of its digits be equal to 82?

b) Can the quotient of this number and the sum of its digits be equal to 83?

c) What is the greatest natural value that the quotient of a given number and the sum of its digits can have?

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The country of Delphinia has the following income tax system (the monetary unit of Delphinia is gold):

a) The two brothers earned a total of 1000 gold. How can they best distribute this money among themselves so that as much money as possible remains in the family after taxes? When dividing, everyone receives an integer number of gold pieces.

b) What is the most profitable way to distribute the same 1000 gold pieces among three brothers, provided that each also receives a whole number of gold pieces?

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Petya multiplied some natural number by a neighboring natural number, and received a product equal to A. Vasya multiplied some even natural number by a neighboring even natural number and obtained a product equal to b.

a) Can the modulus of the difference of numbers a And b equal 8?

b) Can the modulus of the difference of numbers a And b equal 11?

c) What values can the modulus of the difference of numbers take? a And b?

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Find all pairs of natural numbers and such that if you add the decimal notation of the number to the right to the decimal notation of the number, you get a number greater than the product of the numbers and by

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There are more than 40 but less than 48 integers written on the board. The arithmetic mean of these numbers is −3, the arithmetic mean of all positive ones is 4, and the arithmetic mean of all negative ones is −8.

a) How many numbers are written on the board?

b) Which numbers are written more: positive or negative?

c) What is the largest number of positive numbers that can be among them?

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There are stone blocks: 50 pieces of 800 kg each, 60 pieces of 1,000 kg each and 60 pieces of 1,500 kg each (the blocks cannot be split).

a) Is it possible to transport all these blocks simultaneously on 60 trucks, each with a carrying capacity of 5 tons, assuming that the selected blocks will fit into the truck?

b) Is it possible to transport all these blocks simultaneously on 38 trucks, each with a carrying capacity of 5 tons, assuming that the selected blocks will fit into the truck?

c) What is the smallest number of trucks, each with a carrying capacity of 5 tons, will be needed to remove all these blocks at the same time, assuming that the selected blocks will fit in the truck?

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Given a three-digit natural number (the number cannot start from zero), not a multiple of 100.

a) Can the quotient of this number and the sum of its digits be equal to 90?

b) Can the quotient of this number and the sum of its digits be equal to 88?

c) What is the greatest natural value that the quotient of a given number and the sum of its digits can have?

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a) How many numbers are written on the board?

b) Which numbers are written more: positive or negative?

c) What is the largest number of positive numbers that can be among them?

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Are given n various natural numbers that make up an arithmetic progression

a) Can the sum of all these numbers be equal to 14?

b) What is the largest value n, if the sum of all given numbers is less than 900?

c) Find all possible values n, if the sum of all given numbers is 123.

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Can you give an example of five different natural numbers whose product is 1512 and

b) four;

do they form a geometric progression?

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Find all prime numbers for each of which there is an integer such that the fraction can be reduced by

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Given a sequence of natural numbers, each next term differs from the previous one by either 10 or 6 times. The sum of all terms of the sequence is 257.

a) What is the smallest number of terms this sequence can have?

b) What is the largest number of terms this sequence can have?

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One of the most charismatic and prominent artists of Russian cinema has recently seemed to disappear from the public eye. So little is heard about Alexander Domogarov that his many fans might think that the actor has closed himself off from the world. However, he regularly reminds himself of himself on social networks, where an alarming post appeared a few hours ago.

Let us remind you that the 53-year-old People's Artist of Russia, in addition to filming films, plays in the theater with pleasure and pride. Since 1995, Domogarov has served in the capital's Mossovet Theater, where he performed roles in many performances, three of which are in the current repertoire. The actor is considered the star of this theater, photographs of Domogarov on stage decorate the entrance, many fans go to see performances with his participation.

But in his publication in Alexander Yuryevich he said that he was “removed from the performances” and “this is very serious.”

Removed from performances! So bear it! I feel calmer than walking in and saying hello to “colleagues” who spit in the back! - writes the artist. - I will no longer allow you to fire and appoint, remove and return, give on tour or not, for some reason. ...But when I was removed from all the performances, to the delight of my “colleagues,” a statement was written. Written on January 9th. It hasn't been signed yet. But, dear colleagues, it will be signed, even purely legally. All our agreements with the theater will be fulfilled on my part, so sometimes you will have to tolerate me “colleagues” when I have to pick up my things in the dressing room, and in the future the theater will forget, just as you forgot the performances that lasted 10-12 years , collecting halls, and you will forget how you destroyed them. Live, God is your judge. Goodbye "colleagues".

We reached Alexander Domogarov with a request to comment on the situation.

Don't read my posts, because there is some truth in them and only some. But in principle, it corresponds to reality,” Alexander Domogarov answered and hung up.

Let us remember that Alexander Domogarov was officially married three times. His first wife, Natalya Sagoyan, gave birth to his son Dmitry. 10 years ago, the actor’s first-born died in an accident. From his second wife Irina Gunenkova, the actor has a son, Alexander Domogarov, who also became an actor. The third wife, actress Natalya Gromushkina, lived with him in marriage for 4 years. Three years ago, the actor said in: “My son was killed in a car accident, I didn’t find any closure, but I didn’t get angry with the Country! It’s like this all over the world - there are strong and there are invulnerable. But I will and do solve my problem myself. And I will solve it, and I will not yap at the strength and power of those in power. I will decide and decide. And the country gives me this opportunity.”