Number theory

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Problem: Find a six-digit number, the products of which by \( n(n=2,3,4,5,6) \) give numbers in arbitrary order, obtained from the desired by the principle of circular replacement.

12.2.1 Number theory

3.81 $

Problem: Find all 4-digit numbers \( a b c d \) (where \( a, b, c, d \) are decimal digits), each of which is a divisor of at least one of the three four-digit numbers \( b c d a, c d a b \), dabc composed of it.

12.2.2 Number theory

7.62 $

Problem: Let \( m, n \) be numbers greater than 1 . Prove that \( m^{n} \) is presented as a sum of consecutive odd numbers.

12.2.3 Number theory

3.05 $

Problem: One stone was put in each of the points \( n \) with the cooridinates \( x=1, x=2, \ldots, x=n \) in ascending order of their weight. The weight of the lightest stone is \( 3 \mathrm{~kg} \). The weight of each next stone is \( 1 \mathrm{~kg} \) less, than the twice weight of the previous one. Find the total weight of the first 10 stones. For what \( n(9

12.2.4 Number theory

7.62 $

Problem: Let \( n \) be a positive even integer and \( a, b \) are positive coprime integers. Find \( a, b \) if \( a+b \) divides \( a^{n}+b^{n} \).

12.2.5 Number theory

2.54 $

Problem: Find all positive integers \( n \), for which the number, obtained by erasing the last digit of \( n \) is a divisor of \( n \).

12.2.6 Number theory

2.54 $

Problem: Determine all integers \( x \), for which \( \frac{x^{3}-3 x+2}{2 x+1} \) is an integer.

12.2.7 Number theory

3.81 $

Problem: Let \( (a, b)=1 \). Prove that \( (a+b, a-b)=1 \) or 2 .

12.2.8 Number theory

0 $

Problem: The sum of two positive numbers is equal to 5432 , and their least common multiple is 223020 . Find these numbers.

12.2.9 Number theory

3.81 $

Problem: Let \( (a, b)=10 \). Find all possible values of \( \left(a^{3}, b^{4}\right) \).

12.2.10 Number theory

3.05 $

Problem: Let's prove that \( n !+1 \) and \( (n+1) !+1 \) are coprime.

12.2.11 Number theory

3.81 $

Problem: What is the remainder of the division of \( x_{2018}+ \) \( +x_{2019} \) by 9 , if \( x_{1}=x_{2}=1, x_{2 n+1}=2 x_{2 n}+1 \), \( x_{2 n+2}=x_{1}++x_{2}+\cdots+x_{2 n+1} \).

12.2.12 Number theory

3.81 $

Problem: Find the solution of the comparison: \[ x^{8} \equiv 62(\bmod 169) \text {. } \]

12.2.14 Number theory

3.05 $

Problem: Find the solution of the comparison: \[ 355 x \equiv 257 \cdot 156(\bmod 338) . \]

12.2.13 Number theory

3.05 $

Problem: Find all integers \( m \), for which the fraction can be reduced: \[ \frac{m^{3}-5 m^{2}+5 m-2}{m^{2}-3 m+1} \text {. } \]

12.2.15 Number theory

2.54 $

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