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#### Divisibility of a number. General properties

Do you think that among the four consecutive natural numbers there will be at least one that is divisible a) by 2? b) by 3? c) by 4? d) by 5?

#### Divisibility of a number. General properties , Examples and counterexamples. Constructive proofs , Integer and fractional parts. Archimedean property

Does there exist a number h such that for any natural number n the number [$h \times 1969^n$] is not divisible by [$h \times 1969^{n-1}$]?

#### Divisibility of a number. General properties , Lines and planes in space , Pigeonhole principle (other)

Out of the given numbers 1, 2, 3, …, 1000, find the largest number m that has this property: no matter which m of these numbers you delete, among the remaining 1000 – m numbers there are two, of which one is divisible by the other.

#### Divisibility of a number. General properties , Pigeonhole principle (other) , Proof by contradiction

We are given 111 different natural numbers that do not exceed 500. Could it be that for each of these numbers, its last digit coincides with the last digit of the sum of all of the remaining numbers?

#### Divisibility of a number. General properties , Integer and fractional parts. Archimedean property

Prove that for a real positive α and a positive integer d, [α / d] = [[α] / d] is always satisfied.

#### Divisibility of a number. General properties , Partitions into pairs and groups bijections , Pigeonhole principle (other)

From the set of numbers 1 to 2n, n + 1 numbers are chosen. Prove that among the chosen numbers there are two, one of which is divisible by another.

#### Divisibility of a number. General properties

Write out in a row the numbers from 1 to 9 $($every number once$)$ so that every two consecutive numbers give a two-digit number that is divisible by 7 or by 13.

#### Divisibility of a number. General properties

In a graph, three edges emerge from each vertex. Can there be a 1990 edges in this graph?

#### Divisibility of a number. General properties , Fermat's little theorem , Pigeonhole principle (other)

Let p be a prime number, and a is not divisible by p. Prove that there is a positive integer b such that ab $\equiv$ 1 $($mod p$)$.

#### Divisibility of a number. General properties

Can there be exactly 100 roads in a state in which three roads leave each city?

#### Divisibility of a number. General properties , Partitions into pairs and groups bijections , Pigeonhole principle (other)

You are given 11 different natural numbers that are less than or equal to 20. Prove that it is always possible to choose two numbers where one is divisible by the other.

#### Divisibility of a number. General properties , Linear recurrent relations , Number sequences (other)

The sequence of numbers $a_1$, $a_2$, … is given by the conditions $a_1$ = 1, $a_2$ = 143 and for all n $≥$ 2.
Prove that all members of the sequence are integers.

#### Divisibility of a number. General properties , Examples and counterexamples. Constructive proofs , Number sequences (other)

Is there a sequence of natural numbers in which every natural number occurs exactly once, and for any k = 1, 2, 3, … the sum of the first k terms of the sequence is divisible by k?

#### Boundedness, monotonicity , Divisibility of a number. General properties , Examples and counterexamples. Constructive proofs , Identical transformations , Sequnces

Prove that for any natural number $a_1> 1$ there exists an increasing sequence of natural numbers $a_1, a_2, a_3$, …, for which $a_1^2+ a_2^2 +…+ a_k^2$ is divisible by $a_1+ a_2+…+ a_k$ for all k ≥ 1.

#### Decimal number system , Divisibility of a number. General properties , Set cardinality. One-to-one correspondence

Which five-digit numbers are there more of: ones that are not divisible by 5 or those with neither the first nor the second digit on the left being a five?

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