It is known that 45 numbers and by 7. Divisibility of integers and remainders - write off antoshka

This statement is a sign of divisibility by numbers, which can be represented as a product of two coprime numbers.

For example, since 6 = 2 ∙ 3 and D (2, 3) = 1, we obtain a sign of divisibility by 6. In order for a natural number to be divisible by 6, it is necessary and sufficient that it be divisible by both 2 and 3 .

Note that this feature can be used multiple times.

c) Private, obtained by dividing two given numbers and
their greatest common divisor are coprime
numbers.

This property can be used when checking the correctness of the found greatest common divisor of given numbers. For example, let's check whether the number 12 is the greatest common divisor of the numbers 24 and 36. To do this, according to the last statement, we divide 24 and 36 by 12. We get the numbers 2 and 3, respectively, which are coprime. Consequently,

D(24, 36) = 12.

Exercises

1. The numbers 36 and 45 are given.

a) Find all the common divisors of these numbers.

b) Can you name all their common multiples?

c) Find three three-digit numbers that are common multiples of the given numbers.

d) What are D(36, 45) and K(36, 45)? How to check the correctness of the received answers?

2. Are the entries correct:

a) D(32, 8) = 8 and K(32,8) = 32;

b) D(17.35)=1 and K(17.35)=595;

c) D(255,306) = 17 and K(255,306),= 78030,

3. Find K(a, b) if it is known that:

a) a = 47,b=105 and D(47,105)= 1;

b) a = 315, b = 385 and D (315.385) = 35.

4. Formulate signs of divisibility by 12,15,18,36,45,75.

5. From the set of numbers 1032, 2964,5604,8910, 7008 write out those that are divisible by 12.

6. Are 548 and 942 divisible by 18?

7. To the number 15, add on the left and right; one digit so that the resulting number is divisible by 15.

8. Find the numbers a and 6 of the number 72, if it is known that this number is divisible by 45.

9 Without multiplying and dividing by a corner, determine which of the following products are divisible by 30:

a) 105∙20; 6)47∙12∙5; c) 85∙33∙7.

10. Without performing addition or subtraction, determine which expressions are divisible by 36.

a) 72 + 180 + 252; c) 180 + 252 + 100;

b) 612-432; d) 180 + 250 + 200.

91. Prime Numbers

Prime numbers play a big role in mathematics - in essence, they are the "bricks" from which composite beginnings are built. This is stated in a theorem called the fundamental theorem of natural number arithmetic, which is given without proof:

Theorem: Any composite number can be uniquely represented as a product of prime factors.

For example, the notation 110 = 2∙5∙11 is a representation of the number 110 as a product of prime factors or its decomposition into prime factors.

Two decompositions of a number into prime factors are considered the same if they differ from each other only in the order of the factors. Therefore, the representation of the number 110 as a product of 2∙5∙11 or a product of 5∙2∙11 is, in essence, the same factorization of the number 110 into prime factors.

When decomposing numbers into prime factors, they use the signs of divisibility by 2, 3, 5, etc. Let us recall one of the ways to write the decomposition of numbers into prime factors. Let us factorize, for example, the number 90. The number 90 is divisible by 2. Hence, 2 is one of the prime factors in the decomposition of the number 90. Divide 90 by 2. We write the number 2 to the right of the equal sign, and the quotient 45 under the number 90. The number We divide 45 by a prime number 3, we get 15. We divide 15 by 3, we get 5. The number 5 is prime, when we divide it by 5 we get 1. The factorization is completed.

90 = 2∙3∙3∙5

When decomposing a number into prime factors, the product of identical factors is represented as a power: 90 = 2∙3 2∙5; 60 = 2 2 ∙3∙5; 72 = 2 3 ∙ 3 2 . Such a decomposition of a number into prime factors is called canonical.

In connection with the possibility of representing any composite number as a product of prime factors, it becomes necessary to determine whether a given number is prime or composite. Ancient Greek mathematicians, who knew many properties of prime numbers, were already able to solve this problem. So, Eratosthenes (III century BC) invented a method for obtaining prime numbers not exceeding the natural number a. Let's use it to find all prime numbers up to 50.

We write down all natural numbers from 1 to 50 and cross out the number 1 - it is not prime. The number 2 is prime, circle it. After that, we cross out every second number after 2, i.e. numbers 4,6,8,...

The first non-crossed out number 3 is prime, circle it. And cross out every third number after 3, i.e. numbers 9, 15, ... (numbers 6,12, etc. are crossed out earlier).

The first not crossed out number 5 is prime, we will also circle it. Cross out every fifth number after 5, etc.

1 23 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

Those numbers that remain after four deletions (excluding the numbers 2,3,5 and 7) are not divisible by either 2, or 3, or 5, or 7. In arithmetic, it is proved that if a natural number a is greater than one , is not divisible by any of the prime numbers whose square does not exceed o, then a the number is prime. Since 7 2 = 49 and 49< 50, то все оставшиеся числа - простые.

So the prime numbers not exceeding 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

The described method of obtaining prime numbers is called the sieve of Eratosthenes, as it allows you to sift out composite numbers one by one.

Using the method proposed by Eratosthenes, one can find all prime numbers that do not exceed a given number a. But he does not answer the question whether the set of primes is finite or not, because it could turn out that all numbers, starting from some number, are composite and the set of primes is finite. Another Greek mathematician, Euclid, dealt with this problem. He proved that the set of prime numbers is infinite.

Indeed, suppose that the set of primes is finite and exhausted by the numbers 2, 3, 5, 7, 7 - the largest prime number. We multiply all prime numbers and denote their product by a. Let's add 1 to this number. What will be the resulting number

and + 1 - simple or compound?

prime number a+1 cannot be, because it is greater than the largest prime number, and by assumption such numbers do not exist. But it cannot be composite either: if a+ 1 .composite, then it must have at least one prime divisor q. Since the number

a = 2∙3∙5∙...∙ R is also divisible by this prime number q, then the difference ( a + 1) - a, i.e. the number 1 is divisible by q, which is impossible.

So, the number a is neither prime nor composite, but this cannot be either - any number other than 1 is either prime or composite. Therefore, our proposition that the set of primes is finite and is the largest prime number is false, and hence the set of primes is infinite.

Exercises

1. From the set of numbers 13, 27, 29, 51, 67 write out simple
numbers, and factorize the composites into prime factors.

2. Prove that the number 819 is not a prime number.

3. Factorize the numbers 124.588.2700.3780 into prime factors.

4. What number has a decomposition:

a) 2 3 ∙ 3 2 7 ∙ 13; b) 2 2 ∙ 3 ∙ 5 3 ?

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

Recall that the 53-year-old National artist Russia, in addition to filming a movie, plays with pleasure and pride in the theater. Since 1995, Domogarov has been serving in the capital's Mossovet Theater, where he has played roles in many performances, three of which are in the current repertoire. The actor is considered the star of this theater, photographs with Domogarov on stage adorn the entrance, many fans go to performances with his participation.

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

Removed from performances! So endure! I feel calmer than walking in and saying hello to "colleagues" who spit in the back! - writes the artist. - I will no longer allow, for some desire, to dismiss and appoint, remove and return, give on tour or not give. ... But as soon as I was removed from all the performances, to the delight of my "colleagues", a statement was written. Written January 9th. It has not yet been signed. 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 suffer me "colleagues" when I have to pick up my things in the dressing room, and in the future the theater will forget, just like you forgot the performances that went on for 10-12 years , collecting the halls, and you will forget how you destroyed them. Live, God is your judge. Goodbye colleagues.

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

You do not read my posts, because there is some truth in them and only a fraction. But in principle, it corresponds to reality, - Alexander Domogarov answered and hung up.

Recall that Alexander Domogarov was officially married three times. The first wife Natalya Sagoyan gave birth to his son Dmitry. 10 years ago, the first-born actor died in an accident. From his second wife, Irina Gunenkova, the actor has a son, Alexander Domogarov, he also became an actor. The third wife, actress Natalya Gromushkina, was married to him for 4 years. Three years ago, the actor said in: “My son was killed in a car accident, I didn’t find the ends, but I didn’t get mad at the Country! All over the world so - there are strong and there are invulnerable. But I will solve and solve my problem myself. And I will solve it, but I will not yelp at the power and those in power. I will decide and decide. And the country gives me this opportunity.”

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Numbers are written in a row: , , ..., , The signs “+” and “-” are randomly placed between them and the resulting sum is found.

Can 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 is 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 (by 2, by 3, etc.) are written out 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 rest of the numbers are n, are erased. For example, if the numbers 1, 3, 3, 4 are conceived, then the set 1, 3, 4, 5, 6, 7, 8, 10, 11 will be written on the board.

a) Give an example of conceived 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 will be written on the board?

c) Give all the examples of the intended 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 is also a natural number.

a) What highest value can take the area of a rectangle?

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

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 clean sides, each of the numbers 1, -2, -3, 4, -5, 7, -8, 9 is rewritten one at a time. After that, the numbers on each card are added up, and the resulting eight sums are multiplied.

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Several integers are conceived. The set of these numbers and all their possible sums (by 2, by 3, etc.) are written out on the board in non-decreasing order. For example, if the numbers 2, 3, 5 are conceived, 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 conceived?

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

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

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

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There are 8 cards. Each of the numbers is written on them one at a time:

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

−11, 12, 13, −14, −15, 17, −18, 19.

After that, the numbers on each card are added up, and the resulting eight amounts are multiplied.

a) Can the result be 0?

b) Can 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, different from zero) is given, made up of natural numbers whose decimal notation does not contain the digit 9.

(a) Can there be ten terms in such a progression?

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

c) Prove that the number of terms of any such progression is at most 72.

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

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Each of the numbers 1, -2, -3, 4, -5, 7, -8, 9 is written one by one on 8 cards. The cards are turned over and shuffled. On their clean sides, each of the numbers 1, -2, -3, 4, -5, 7, -8, 9 is written again one at a time. After that, the numbers on each card are added up, 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

succeed?

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

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

b) Can the sum of all the numbers on the board be 63 at some point?

c) What is the shortest time it takes for the number 784 to 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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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, the modulus of the difference between the sums found is found and the resulting 6 numbers are added.

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 arbitrarily put a plus or minus sign, after which each of the formed numbers of the second set is added to each of the formed numbers of the first set, and then all 45 results are added. What is the smallest modulo sum and what is the largest sum that can be obtained as a result?

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

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

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

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

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

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

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 that stood through one, they found the difference between the larger and smaller ones. For what is the largest integer k you can arrange the numbers in such a way that all the differences are not less than k?

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

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

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

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n n n

a) Give an example of conceived 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 will be written on the board?

c) Give all the examples of the intended 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 distinct numbers such that they can be represented as 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 conceived 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 will 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 some natural number by a neighboring natural number, and got 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 to 6?

m and n equal 13?

m and n?

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

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Each of the group of students went to the cinema or to the theater, while it is possible that one 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 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 the group?

c) What was the smallest proportion of girls in 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 (a number cannot start from zero) that is 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 largest natural value of a given number and the sum of its digits?

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The country of Delphinia has the following income tax system ( currency unit Dolphins - golden):

a) The two brothers earned a total of 1000 gold. How is it most profitable for them to distribute this money among themselves, so that the family has as much as possible more money after tax? When dividing, everyone receives an integer number of gold pieces.

b) What is the best way to distribute the same 1000 gold pieces among the three brothers, provided that each also receives an integer number of gold coins?

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

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

b) Can the module of the difference of numbers a and b equal to 11?

c) What values \u200b\u200bof the modulus of the difference of numbers can take a and b?

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

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

a) How many numbers are written on the board?

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

c) What is the greatest number of positive numbers among them?

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

a) Is it possible to take away all these blocks at the same time on 60 trucks, with a carrying capacity of 5 tons each, assuming that the chosen blocks will fit in the truck?

b) Is it possible to take away all these blocks at the same time on 38 trucks, with a carrying capacity of 5 tons each, assuming that the chosen blocks will fit in the truck?

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

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Given a three-digit natural number (a number cannot start from zero) that is 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 largest natural value of a given number and the sum of its digits?

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a) How many numbers are written on the board?

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

c) What is the greatest number of positive numbers among them?

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

a) Can the sum of all given 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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Is it possible to give an example of five different natural numbers whose product is equal to 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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A sequence of natural numbers is given, and each next term differs from the previous one either by 10 or 6 times. The sum of all terms in the sequence is 257.

a) What is the smallest number of terms that can be in this sequence?

b) What is the maximum number of members that can be in this sequence?

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If two numbers a and b when dividing by a number m give the same remainder, then we say that a is congruent to b modulo m. Write it down like this a ≡ b (mod m)

If a a > b, then the greatest common divisor a and b is equal to the greatest common divisor a-b and b.

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: We cross out from 999 numbers less than 1000 numbers that are multiples of 5: there are 199 of them (999/5 = 199). Next, we cross out the 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 simultaneously multiples of 5; they will be crossed out twice. In total, we should cross out 199 + 142 - 28 = 313 numbers.

It remains 999 - 313 = 686. Answer: 686 numbers.

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

The solution of the problem

Given that 2009⋮7, then the remainder will be 2012 2 ≡ 3 2 ≡ 2(mod7)

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

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, when divided by 8, leave a remainder of 7.

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 when divided by 8, one of the three numbers 0, 1, 4. 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 satisfied: either one of the squares, or all three, when divided by 8, have odd remainders.

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

5. Are there natural numbers 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 that n 2 + n + 1 is always odd (this could also be seen 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?

I way. When writing out three-digit numbers divisible by 11, you can find three numbers among them, in the recording of which all numbers from 0 to 9 participate. For example, 275, 396.418. With their help, you can make a ten-digit number divisible by 11. For example:

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

II way. To find the required number, we use the criterion of divisibility by 11, according to which the numbers n = a 1 a 2 a 3 ... a 10 (in this case and i are not factors, but digits in the notation of the number n) and S (n) \u003d 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 the "+" sign, B be the sum of digits included in S(n) with the "-" sign. The number A-B, according to the condition of the problem, must be divisible by 11. Let's put B - A = 11, in addition, obviously, A + B = 1 + 2 + 3 + ... + 9 = 45. Solving the resulting system B - A \u003d 11 , A + B \u003d 45, we find, A \u003d 17, B \u003d 28. Let's select a group of five different numbers with a sum of 17. For example, 1 + 2 + 3 + 5 + 6 \u003d 17. We will take these numbers as numbers with odd numbers . As digits with even numbers, we take the remaining ones - 4, 7, 8, 9, 0.

We see that the condition of the problem is satisfied, for example, by the number 1427385960.

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

Solution

Consider the remainder when dividing the number 1234 by 25. All numbers less than it give other remainders, since they are their own remainders. The remainder when 1234 is divided by 25 is 9, since 1234=49⋅25+9, this is the answer.

8. Having received a deuce in geography, Vasya decided to break geographical map apart. Each scrap that falls into his hands he tears into four parts. Can he ever get exactly 2012 chunks? 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 like 1+3N, where N is the number of pieces he tore apart. The number 2014 has this form, so it will get 2014 pieces, while others cannot be represented in this form (they have 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 one less 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, the three and five enter the maximum in the first degree, and the two in the second (in the number 4). So the desired number is 60−1 = 59.

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