Found problems: 71
Consider $10$ distinct positive integers that are all prime to each other (that is, there is no a prime factor common to all), but such that any two of them are not prime to each other. What is the smallest number of distinct prime factors that may appear in the product of $10$ numbers?
The sequence ($a_n$) is defined by $a_1 = a_2 = 1$ and $a_{n+2 }= a_{n+1} +a_n +k$, where $k$ is a positive integer.
Find the least $k$ for which $a_{1991}$ and $1991$ are not coprime.
If $n$ is a positive integer and $a, b$ are relatively prime positive integers, calculate $(a + b,a^n + b^n)$.
Call a finite set of positive integers [i]independent [/i] if its elements are pairwise coprime, and [i]nice [/i] if the arithmetic mean of the elements of every non-empty subset of it is an integer.
a) Prove that for any positive integer $n$ there is an $n$-element set of positive integers which is both independent and nice.
b) Is there an infinite set of positive integers whose every independent subset is nice and which has an $n$-element independent subset for every positive integer $n$?
Prove that of any six four-digit numbers, mutual prime in total, you can always choose five numbers that are also relatively prime in total.
[hide=original wording]Докажите, что из любых шести четырехзначных чисел, взаимно простых в совокупности, всегда можно выбратьпя ть чисел, также взаимно простых в совокупности.[/hide]
Prove that there are arithmetic progressions of arbitrary length, consisting of different pairwise coprime natural numbers.
The Fermat-numbers are defined by $F_n = 2^{2^n}+1$ for $n\in N$.
(a) Prove that $F_n = F_{n-1}F_{n-2}....F_1F_0 +2$ for $n > 0$.
(b) Prove that any two different Fermat numbers are coprime
Prove that among every ten consecutive natural numbers there is one that is coprime to each of the other nine.
Determine all integers $n \ge 2$ such that $n \mid s_n-t_n$ where $s_n$ is the sum of all the integers in the interval $[1,n]$ that are mutually prime to $n$, and $t_n$ is the sum of the remaining integers in the same interval.
Given an integer $n \ge 2$, evaluate $\Sigma \frac{1}{pq}$ ,where the summation is over all coprime integers $p$ and $q$ such that $1 \le p < q \le n$ and $p + q > n$.
For positive integer $n$, find all pairs of coprime integers $p$ and $q$ such that $p+q^2=(n^2+1)p^2+q$
Prove that if the numbers $ x_1 $ and $ x_2 $ are roots of the equation $ x^2 + px - 1 = 0 $, where $ p $ is an odd number, then for every natural $n$number $ x_1^n + x_2^n $ and $ x_1^{n+1} + x_2^{n+1} $ are integer and coprime.
$(FRA 1)$ Let $a$ and $b$ be two nonnegative integers. Denote by $H(a, b)$ the set of numbers $n$ of the form $n = pa + qb,$ where $p$ and $q$ are positive integers. Determine $H(a) = H(a, a)$. Prove that if $a \neq b,$ it is enough to know all the sets $H(a, b)$ for coprime numbers $a, b$ in order to know all the sets $H(a, b)$. Prove that in the case of coprime numbers $a$ and $b, H(a, b)$ contains all numbers greater than or equal to $\omega = (a - 1)(b -1)$ and also $\frac{\omega}{2}$ numbers smaller than $\omega$
The sequences $(a_n)$ and $(b_n)$ are defined by $a_1 = b_1 = 1$ and $a_{n+1} = a_n +b_n, b_{n+1} = a_nb_n$ for $n = 1,2,...$ Show that every two distinct terms of the sequence $(a_n)$ are coprime
Given any integer $n \ge 2$, show that there exists a set of $n$ pairwise coprime composite integers in arithmetic progression.
Call a positive integer $N \ge 2$ [i]special [/i] if for every k such that $2 \le k \le N, N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). Find all special positive integers.
For a positive integer $n$, let $\alpha(n)$ be the arithmetic mean of the divisors of $n$, and let $\beta(n)$ be the arithmetic mean of the numbers $k \le n$ with $\text{gcd}(k,n)=1$.
Determine all positive integers $n$ with $\alpha(n)=\beta(n)$.
Let $m$ and $n$ be distinct positive integers. Prove that $2^{2^m} + 1$ and $2^{2^n} + 1$ have no common divisor greater than $1$.
Does there exist an even positive integer $n$ for which $n+1$ is divisible by $5$ and the two numbers $2^n + n$ and $2^n -1$ are co-prime?
Suppose $S = \{a_1, a_2,..., a_{15}\}$ is a set of $1 5$ distinct positive integers chosen from $2 , 3, ... , 2012$ such that every two of them are coprime. Prove that $S$ contains a prime number.
(Note: Two positive integers $m, n$ are coprime if their only common factor is 1)
Let $a, b, c$ be nonzero integers, with $1$ as their only positive common divisor, such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}= 0$. Find the number of such triples $(a, b, c)$ with $50 \ge |a| \ge |b| \ge |c| 1$.
Prove that from an arbitrary set of three-digit numbers, including at least four numbers that are mutually prime, you can choose four numbers that are also mutually prime
Find all elements $n \in A = \{2,3,...,2016\} \subset \mathbb{N}$ such that:
every number $m \in A$ smaller than $n$, and coprime with $n$, must be a prime number
The natural numbers $a$ and $b$ satis fy the inequalities $a > b > 1$ . It is also known that the equation
$\frac{a^x - 1}{a - 1}=\frac{b^y - 1}{b - 1}$ has at least two solutions in natural numbers, when $x > 1$ and $y > 1$.
Prove that the numbers $a$ and $b$ are coprime (their greatest common divisor is $1$).
Given natural numbers $a,b$ and function $f: \mathbb{N} \to \mathbb{N} $ such that for any natural number $n, f\left( n+a \right)$ is divided by $f\left( {\left[ {\sqrt n } \right] + b} \right)$. Prove that for any natural $n$ exist $n$ pairwise distinct and pairwise relatively prime natural numbers ${{a}_{1}}$, ${{a}_{2}}$, $\ldots$, ${{a}_{n}}$ such that the number $f\left( {{a}_{i+1}} \right)$ is divided by $f\left( {{a}_{i}} \right)$ for each $i=1,2, \dots ,n-1$ .
(Here $[x]$ is the integer part of number $x$, that is, the largest integer not exceeding $x$.)