Let $\mathbb N$ denote set of all natural numbers and let $f:\mathbb{N}\to\mathbb{N}$ be a function such that $\text{(a)} f(mn)=f(m).f(n)$ for all $m,n \in\mathbb{N}$; $\text{(b)} m+n$ divides $f(m)+f(n)$ for all $m,n\in \mathbb N$. Prove that, there exists an odd natural number $k$ such that $f(n)= n^k$ for all $n$ in $\mathbb{N}$.

Let $n$ be a natural number, $n > 2$. Prove that if $\frac{b^n-1}{b-1}$ is a prime power for some positive integer $b$ then $n$ is prime.

For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$ Let $n$ be a positive integer. Prove that the following conditions are equivalent: (i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$, (ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.

Some positive integer $n$ is written on the board. Andrey and Petya is playing the following game. Andrey finds all the different representations of the number n as a product of powers of prime numbers (values degrees greater than 1), in which each factor is greater than all previous or equal to the previous one. Petya finds all different representations of the number $n$ as a product of integers greater than $1$, in which each factor is divisible by all the previous factors. The one who finds more performances wins, if everyone finds the same number of representations, the game ends in a draw. Find all positive integers $n$ for which the game will end in a draw. Note. The representation of the number $n$ as a product is also considered a representation consisting of a single factor $n$.

Solve $2a^2+3a-44=3p^n$ in positive integers where $p$ is a prime.

Some positive integer $n$ is written on the board. Andrey and Petya is playing the following game. Andrey finds all the different representations of the number n as a product of powers of prime numbers (values degrees greater than 1), in which each factor is greater than all previous or equal to the previous one. Petya finds all different representations of the number $n$ as a product of integers greater than $1$, in which each factor is divisible by all the previous factors. The one who finds more performances wins, if everyone finds the same number of representations, the game ends in a draw. Find all positive integers $n$ for which the game will end in a draw. Note. The representation of the number $n$ as a product is also considered a representation consisting of a single factor $n$.

Let $n$ be a natural number, $n > 2$. Prove that if $\frac{b^n-1}{b-1}$ is a prime power for some positive integer $b$ then $n$ is prime.

For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$ Let $n$ be a positive integer. Prove that the following conditions are equivalent: (i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$, (ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.

For what pairs of natural numbers $(a, b)$ is the expression $$(a^6 + 21a^4b^2 + 35a^2b^4 + 7b^6) (b^6 + 21b^4a^2 + 35b^2a^4 + 7a^6)$$ the power of a prime number?

Find all prime numbers whose sixth power does not give remainder $1$ when dividing by $504$

Define $f (n) = n + 1$ if $n = p^k > 1$ is a power of a prime number, and $f (n) =p_1^{k_1}+... + p_r^{k_r}$ for natural numbers $n = p_1^{k_1}... p_r^{k_r}$ ($r > 1, k_i > 0$). Given $m > 1$, we construct the sequence $a_0 = m, a_{j+1} = f (a_j)$ for $j \ge 0$ and denote by $g(m)$ the smallest term in this sequence. For each $m > 1$, determine $g(m)$.