Number theory/Number theory

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Problem: Solve the equation in natural numbers: \[ 6 x^{2}-y^{2}=5 z^{2} \text {. } \]

12.2.16 Number theory

9.91 $

Problem: Is the number \( 2015 \cdot 2016 \cdot 2017 \cdot 2018+1 \) prime?

12.2.17 Number theory

1.49 $

Problem: The set \( A \) contains all and only those natural numbers, the sum of the digits of each of which does not change as a result of multiplication by 11 . Before each number from the set \( A \) all its natural divisors were written in ascending order. Find the largest value among all the numbers written in the third place in any line, and if the desired maximum does not exist, then indicate the word no as an answer.

12.2.18 Number theory

2.97 $

Problem: Find the GCD \( \left(4 \cdot 3^{25}-8^{15} ; 2 \cdot 3^{17}+8^{10}\right) \).

12.2.19 Number theory

3.72 $

Problem: Find the GCD \( \left(3 n^{2}-n+7 ; 4 n^{2}-5\right) \) depending on \( n \).

12.2.20 Number theory

3.72 $

Problem: For which natural numbers \( k \), greater than 50 , but smaller than 100 , there is a number, that is the sum of \( k \) consecutive natural numbers, but is not the sum of \( m \) consecutive natural numbers for any \( m \) from 2 to \( k-1 \) ?

12.2.21 Number theory

9.91 $

Problem: Calculate: \( \left(\left(3^{14}+15^{16}\right)^{17}+1\right)^{18}(\bmod 7) \).

12.2.22 Number theory

1.98 $

Problem: Let \( a, b \) are integers, prove that \( 19 a+9 b \) is divided by 7 only when \( a-b \) is divided by 7 .

12.2.23 Number theory

1.73 $

Problem: We have a sequence of six strictly increasing such positive integers, that starting from the second one, each is a multiple of the previous one, and the sum of all six numbers is 79 . What is the largest number in the sequence?

12.2.24 Number theory

2.97 $

Problem: Find the greatest natural number \( n \), for which \( \left(3^{1024}-1\right) \vdots 2^{n} \).

12.2.25 Number theory

3.22 $

Problem: Prove that there are infinitely many such even positive numbers \( k \), that for every prime \( p \) the number \( p^{2}+k \) is composite.

12.2.26 Number theory

4.95 $

Problem: An increasing sequence of prime numbers \( n_{1}< \) \(

12.2.27 Number theory

3.72 $

Problem: Let's consider the sequence \( a_{1}, a_{2} \ldots \) defined as follows: \( a_{n}=2^{n}-3^{n}+6^{n}-1, n \in \mathbb{N} \). Determine all positive integers, relatively prime to each member of the sequence.

12.2.28 Number theory

3.22 $

Problem: Let \( \overline{x y} \) and \( \overline{y x} \) be two two-digit integers. Prove that their sum is a composite number.

12.2.29 Number theory

0 $

Problem: Find the sum of all such coprime positive integers, that these numbers are divisible by their digits.

12.2.30 Number theory

2.48 $

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