Školjka

%V0 $(FRA 6)$ Consider the integer $d = \frac{a^b-1}{c}$, where $a, b$, and $c$ are positive integers and $c \le a.$ Prove that the set $G$ of integers that are between $1$ and $d$ and relatively prime to $d$ (the number of such integers is denoted by $\phi(d)$) can be partitioned into $n$ subsets, each of which consists of $b$ elements. What can be said about the rational number $\frac{\phi(d)}{b}?$

%V0 $(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$

%V0 $(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$

%V0 Let $a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if $$x^2 - ay^2 - bz^2 + abw^2 = 0$$ has a nontrivial solution in integers, then so does $$x^2 - ay^2 - bz^2 = 0.$$

%V0 Let $m$ be a positive odd integer, $m > 2.$ Find the smallest positive integer $n$ such that $2^{1989}$ divides $m^n - 1.$

%V0 Let $a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^+$ be given and let N$(a_1, a_2, a_3)$ be the number of solutions $(x_1, x_2, x_3)$ of the equation $$\sum^3_{k=1} \frac{a_k}{x_k} = 1.$$ where $x_1, x_2,$ and $x_3$ are positive integers. Prove that $$N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 + ln(2 a_1)).$$