Found problems: 97
For some positive integers $m>n$, the quantities $a=\text{lcm}(m,n)$ and $b=\gcd(m,n)$ satisfy $a=30b$. If $m-n$ divides $a$, then what is the value of $\frac{m+n}{b}$?
[i]Proposed by Andrew Wu[/i]
Let \( n \) be a natural number. The numbers \( 1, 2, \ldots, n \) are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the \( n - 1 \) GCDs obtained? \\
Given an integer $a>1$. Prove that there exists a sequence of positive integers
\[ n_1, n_2, n_3, \ldots \]
Such that
\[ \gcd(a^{n_i+1} + a^{n_i} - 1, \ a^{n_j + 1} + a^{n_j} - 1) =1 \] For every $i \neq j$.
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.
Determine all triples of positive integers $a, b, c$ that satisfy
a) $[a, b] + [a, c] + [b, c] = [a, b, c]$.
b) $[a, b] + [a, c] + [b, c] = [a, b, c] + (a, b, c)$.
Remark: Here $[x, y$] denotes the least common multiple of positive integers $x$ and $y$, and $(x, y)$ denotes their greatest common divisor.
For all positive integers $a$ and $b$, we de ne $a @ b = \frac{a - b}{gcd(a, b)}$ .
Show that for every integer $n > 1$, the following holds:
$n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.
Find all positive integers $m,n$ and prime numbers $p$ for which $\frac{5^m+2^np}{5^m-2^np}$ is a perfect square.
For positive integers $a$ and $b$, $gcd (a, b)$ denote their greatest common divisor and $lcm (a, b)$ their least common multiple. Determine the number of ordered pairs (a,b) of positive integers satisfying the equation $ab + 63 = 20\, lcm (a, b) + 12\, gcd (a,b)$
For any positive integers $n>m$ prove the following inequality:
$$[m,n]+[m+1,n+1]\geq 2m\sqrt{n}$$
As usual, [x,y] denotes the least common multiply of $x,y$
[I]Proposed by A. Golovanov[/i]
Compute gcd $\left( \frac{135^{90}-45^{90}}{90^2} , 90^2 \right)$
Let $14$ integer numbers are given. Let Hamza writes on the paper the greatest common divisor for each pair of numbers. It occurs that the difference between the biggest and smallest numbers written on the paper is less than $91$. Prove that not all numbers on the paper are different.
Find all pairs $(n, m)$ of positive integers such that $gcd ((n + 1)^m - n, (n + 1)^{m+3} - n) > 1$.
Do there exist a positive integer $k$ and a non-constant sequence $a_1, a_2, a_3, ...$ of positive integers such that $a_n = gcd(a_{n+k}, a_{n+k+1})$ for all positive integers $n$?
Let $d$ be a positive divisor of a positive integer $m$ and $(a_l), (b_l)$ two arithmetic sequences of positive integers. It is given that $gcd(a_i, b_j) = 1$ and $gcd(a_k, b_n) = m$ for some positive integers $i,j,k,$ and $n$. Prove that there exist positive integers $t$ and $s$ such that $gcd(a_t, b_s) = d$.
Determine all pairs $(x, y)$ of positive integers such that for $d = gcd(x, y)$ the equation $$xyd = x + y + d^2$$
holds.
[i](Walther Janous)[/i]
Let $19$ integer numbers are given. Let Hamza writes on the paper the greatest common divisor for each pair of numbers. It occurs that the difference between the biggest and smallest numbers written on the paper is less than $180$. Prove that not all numbers on the paper are different.
Let $a, b$ and $c$ be natural numbers. Determine the smallest value that the following expression can take:
$$\frac{a}{gcd\,\,(a + b, a - c)}
+
\frac{b}{gcd\,\,(b + c, b - a)}
+
\frac{c}{gcd\,\,(c + a, c - b)}.$$
.
Remark: $gcd \,\, (6, 0) = 6$ and $gcd\,\,(3, -6) = 3$.
There are $1001$ stacks of coins $S_1, S_2, \dots, S_{1001}$. Initially, stack $S_k$ has $k$ coins for each $k = 1,2,\dots,1001$. In an operation, one selects an ordered pair $(i,j)$ of indices $i$ and $j$ satisfying $1 \le i < j \le 1001$ subject to two conditions:
[list]
[*]The stacks $S_i$ and $S_j$ must each have at least $1$ coin.
[*]The ordered pair $(i,j)$ must [i]not[/i] have been selected before.
[/list]
Then, if $S_i$ and $S_j$ have $a$ coins and $b$ coins respectively, one removes $\gcd(a,b)$ coins from each stack.
What is the maximum number of times this operation could be performed?
[i]Galin Totev[/i]
A list of natural numbers is written on a blackboard. The following operation is performed and repeated: choose any two numbers $a, b$, wipe them out and instead write gcd$(a, b)$ and lcm$(a, b)$. Show that the content of the list no longer changed after a certain point in time.
Determine all sets $M$ of natural numbers such that for every two (not necessarily different) elements $a, b$ from $M$ , $$\frac{a + b}{gcd(a, b)}$$ lies in $M$.
Compute gcd$(5^{2547} - 1, 5^{2004} - 1)$.
Let $n$ be an arbitrary positive integer.
(a) For every positive integers $a$ and $b$, show that $gcd(n^a + 1, n^b + 1) \le n^{gcd(a,b)} + 1$.
(b) Show that there exist infinitely many composite pairs ($a, b)$, such that each of them is not a multiply of the other number and equality holds in (a).
Let $n \ge 2$ be an integer and $p_1 < p_2 < ... < p_n$ prime numbers. Prove that there exists an integer $k$ relatively prime with $p_1p_2... p_n$ and such that $gcd (k + p_1p_2...p_i, p_1p_2...p_n) = 1$ for all $i = 1, 2,..., n - 1$.
Malik Talbi
Let $a, b$ be positive integers and $p, q$ be prime numbers for which $p \nmid q - 1$ and $q \mid a^p - b^p$. Prove that $q \mid a - b$.
Initially a number $6$ is written on a blackboard. At $n$-th step an integer $k$ on the blackboard is replaced by $k+gcd(k,n)$. Prove that at each step the number on the blackboard increases either by $1$ or by a prime number.