Let $n_1,n_2,n_3,...,n_{2000}$ be $2000$ positive integers satisfying $n_1<n_2<n_3<...<n_{2000}$. Prove that $$\frac{n_1}{[n_1,n_2]}+\frac{n_1}{[n_2,n_3]}+\frac{n_1}{[n_3,n_4]}+...+\frac{n_1}{[n_{1999},n_{2000}]} \le 1 - \frac{1}{2^{1999}}$$ where $[a, b]$ denotes the least common multiple of $a$ and $b$.

The first of an infinite triangular spreadsheet the line contains one number, the second line contains two numbers, the third line contains three numbers, and so on. In doing so is in any $k$-th row ($k = 1, 2, 3,...$) in the first and last place the number $k$, each other the number in the table is found, however, than in the previous row the least common of the two numbers above it multiple (the adjacent figure shows the first five rows of this table). We choose any two numbers from the table that are not in their row in the first or last place. Prove that one of the selected numbers is divisible by another. [img]https://cdn.artofproblemsolving.com/attachments/3/7/107d8999d9f04777719a0f1b1df418dbe00023.png[/img]

The least common multiple of positive integers $a, b, c$ and $d$ is equal to $a + b + c + d$. Prove that $abcd$ is divisible by at least one of $3$ and $5$. ( V Senderov)

For positive integers $n, k$ with $1 \le k \le n$, define $$L(n, k) = Lcm \,(n, n - 1, n -2, ..., n - k + 1)$$ Let $f(n)$ be the largest value of $k$ such that $L(n, 1) < L(n, 2) < ... < L(n, k)$. Prove that $f(n) < 3\sqrt{n}$ and $f(n) > k$ if $n > k! + k$.

Find all triples $(a, b, c)$ of natural numbers such that the sets $$\{ gcd (a, b), gcd(b, c), gcd(c, a), lcm (a, b), lcm (b, c), lcm (c, a)\}$$ and $$\{2, 3, 5, 30, 60\}$$ are the same. Remark: For example, the sets $\{1, 2013\}$ and $\{1, 1, 2013\}$ are equal.

Determine the number of all ordered triplets of positive integers $(a, b, c)$, which satisfy the equalities: $$[a, b] =1000, [b, c] = 2000, [c, a] =2000.$$ ([x, y]represents the least common multiple of positive integers x,y)

Determine all such pairs pf positive integers $(a, b)$ such that $a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \cdot lcm(a -1, b)$, where $lcm (a, b)$ denotes the smallest common multiple, and $gcd (a, b)$ denotes the greatest common divisor of numbers $a, b$.

Let $L(n)$ denote the least common multiple of $\{1, 2 . . . , n\}$. (i) Prove that there exists a positive integer $k$ such that $L(k) = L(k + 1) = ... = L(k + 2000)$. (ii) Find all $m$ such that $L(m + i) \ne L(m + i + 1)$ for all $i = 0, 1, 2$.

Given an integer $a>1$. Let $p_1 < p_2 <...< p_k$ be all prime divisors of $a$. For each positive integer $n$ we define: $C_0(n) = a^{2n}, C_1(n) =\frac{a^{2n}}{p^2_1}, .... , C_k(n) =\frac{a^{2_n}}{p^2_k}$ $A = a^2 + 1$ $T(n) = A^{C_0(n)} - 1$ $M(n) = LCM(a^{2n+2}, A^{C_1(n)} - 1, ..., A^{C_k(n)} - 1)$ $A_n =\frac{T(n)}{M(n)}$ Prove that the sequence $A_1, A_2, ... $ satisfies the properties: (i) Every number in the sequence is an integer greater than $1$ and has only prime divisors of the form $am + 1$. (ii) Any two different numbers in the sequence are coprime.

We are given a non-infinite sequence $a_1,a_2…a_n$ of natural numbers. While it is possible, on each turn are chosen two arbitrary indexes $i<j$ such that $a_i \nmid a_j$, and then $a_i$ and $a_j$ are changed with their $gcd$ and $lcm$. Prove that this process is non-infinite and the created sequence doesn’t depend on the made choices.

What is the smallest possible value of the least common multiple of $a, b, c, d$ if we know that these four numbers are distinct and $a + b + c + d = 1000$?

$S$ is any sequence of at least $3$ positive integers. A move is to take any $a, b$ in the sequence such that neither divides the other and replace them by gcd $(a,b)$ and lcm $(a,b)$. Show that only finitely many moves are possible and that the final result is independent of the moves made, except possibly for order.

(a) Prove that for any two positive integers a and b the equation $lcm (a, a + 5) = lcm (b, b + 5)$ implies $a = b$. (b) Is it possible that $lcm (a, b) = lcm (a + c, b + c)$ for positive integers $a, b$ and $c$? (A Shapovalov) PS. part (a) for Juniors, both part for Seniors

Denote by $[a, b]$ the least common multiple of $a$ and $b$. Let $n$ be a positive integer such that $[n, n + 1] > [n, n + 2] >...> [n, n + 35]$. Prove that $[n, n + 35] > [n,n + 36]$.

Find all natural numbers $n$ for which the number of all positive divisors of the number lcm $(1,2,..., n)$ is equal to $2^k$ for some non-negative integer $k$.

a) Do there exist positive integers $a$ and $d$ such that $[a, a+d] = [a, a+2d]$? b) Do there exist positive integers $a$ and $d$ such that $[a, a+d] = [a, a+4d]$? Here $[a, b]$ denotes the least common multiple of integers $a, b$.

Find all pairs of natural numbers $(\alpha,\beta)$ for which, if $\delta$ is the greatest common divisor of $\alpha,\beta$, and $\varDelta$ is the least common multiple of $\alpha,\beta$, then \[ \delta + \Delta = 4(\alpha + \beta) + 2021\]

Find all posible pairs of positive integers $x,y$ such that $$\text{lcm}(x,y+3001)=\text{lcm}(y,x+3001)$$

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]

The natural numbers $a_1 <a_2 <.... <a_n\le 2n$ are such that the least common multiple of any two of them is greater than $2n$. Prove that $a_1 >\left[\frac{2n}{3}\right]$.

Let $n \ge 2$ be a positive integer. Suppose $a_1, a_2, \dots, a_n$ are distinct integers. For $k = 1, 2, \dots, n$, let \[ s_k := \prod_{\substack{i \not= k, \\ 1 \le i \le n}} |a_k - a_i|, \] i.e. $s_k$ is the product of all terms of the form $|a_k - a_i|$, where $i \in \{ 1, 2, \dots, n \}$ and $i \not= k$. Find the largest positive integer $M$ such that $M$ divides the least common multiple of $s_1, s_2, \dots, s_n$ for any choices of $a_1, a_2, \dots, a_n$.

Find all positive integer solutions $(a, b)$ to the equation $$\frac{1}{a}+\frac{1}{b}+ \frac{n}{lcm(a,b)}=\frac{1}{gcd(a, b)}$$ for (i) $n = 2007$; (ii) $n = 2010$.

Natural numbers $m$ and $n$ satisfy $$gcd(m,n)+lcm(m,n) = m+n. $$Prove that one of numbers $m,n$ divides the other.

For any $ a\in\mathbb{Z}_{\ge 0} $ make the notation $ a\mathbb{Z}_{\ge 0} =\{ an| n\in\mathbb{Z}_{\ge 0} \} . $ Prove that the following relations are equivalent: $ \text{(1)} a\mathbb{Z}_{\ge 0} \setminus b\mathbb{Z}_{\ge 0}\subset c\mathbb{Z}_{\ge 0} \setminus d\mathbb{Z}_{\ge 0} $ $ \text{(2)} b|a\text{ or } (c|a\text{ and } \text{lcm} (a,b) |\text{lcm} (a,d)) $ [i]Marin Tolosi[/i] and [i]Cosmin Nitu[/i]

Determine all triples of positive integers $a$, $ b$ and $c$ such that their least common multiple is equal to their sum.