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#### Division with remainder , Periodicity and aperiodicity , Probability theory (other) , Proof by exhaustion

A numerical sequence is defined by the following conditions:
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How many complete squares are found among the first members of this sequence, not exceeding 1,000,000?

#### Division with remainder , Periodicity and aperiodicity , Probability theory (other) , Proof by exhaustion

A numerical sequence is defined by the following conditions:

$\\$
Prove that among the terms of this sequence there are an infinite number of complete squares.

#### Division with remainder

Will the quotient or the remainder change if a divided number and the divisor are increased by 3 times?

#### Division with remainder , Proof by exhaustion

Find all of the natural numbers that, when divided by 7, have the same remainder and quotient.

#### Division with remainder , Pigeonhole principle (other)

Prove that amongst the numbers of the form $19991999…19990…0$ – that is 1999 a number of times, followed by a number of 0s – there will be at least one divisible by 2001.

#### Division with remainder , Pigeonhole principle (other)

Prove that in any group of 2001 whole numbers there will be two whose difference is divisible by 2000.

#### Division with remainder , Pigeonhole principle (other) , The greatest common divisor (GCD) and the least common multiplier (LCM). Mutually prime numbers

All of the integers from 1 to 64 are written in an $8 \times 8$ table. Prove that in this case there are two adjacent numbers, the difference between which is not less than 5. $($Numbers that are in cells which share a common side are called adjacent$)$.

#### Division with remainder , Pigeonhole principle (other) , The greatest common divisor (GCD) and the least common multiplier (LCM). Mutually prime numbers

Prove that, for any integer n, among the numbers n, n + 1, n + 2, …, n + 9 there is at least one number that is mutually prime with the other nine numbers.

#### Division with remainder , Periodicity and aperiodicity , Probability theory (other) , Proof by exhaustion

What has a greater value: 300! or $100^{300}$?

#### Decimal number system , Division with remainder , Pigeonhole principle (other) , The greatest common divisor (GCD) and the least common multiplier (LCM). Mutually prime numbers

An infinite sequence of numbers is given. Prove that for any natural number n that is relatively prime with a number 10, you can choose a group of consecutive digits, which when written as a sequence of digits, gives a resulting number written by these digits which is divisible by n.

#### Division with remainder , Euler's theorem , Pigeonhole principle (other) , The greatest common divisor (GCD) and the least common multiplier (LCM). Mutually prime numbers

Prove that for any odd natural number, a, there exists a natural number, b, such that $2^b$ – 1 is divisible by a.

#### Division with remainder , Periodicity and aperiodicity , Probability theory (other) , Proof by exhaustion

Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.

#### Decimal number system , Division with remainder , Pigeonhole principle (other)

Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 2001.

#### Division with remainder , Pigeonhole principle (other)

Prove that if $a, b, c$ are odd numbers, then at least one of the numbers $ab-1, bc-1, ca-1$ is divisible by 4.

#### Decimal number system , Division with remainder

Prove that for any number d, which is not divisible by 2 or by 5, there is a number whose decimal notation contains only ones and which is divisible by d.

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