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 \[ \delta + \Delta = 4(\alpha + \beta) + 2021\]

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

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 $k$.

Find the least positive integer $m$ such that $lcm(15,m) = lcm(42,m)$. Here $lcm(a, b)$ is the least common multiple of $a$ and $b$.

Find a set of positive integers with the greatest possible number of elements such that the least common multiple of all of them is less than $2011$.

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 $a_1, a_2, \dots, a_n$.

Determine all triples of positive integers $(x, y, z)$ that satisfy $$x < y < z, \ \ gcd(x, y) = 6, \ \ gcd(y, z) = 10, \ \ gcd(z, x) = 8 \ \ and \ \ lcm(x, y, z) = 2400.$$

Let $k > 1$ be a fixed integer. Prove that there are infinite positive integers $n$ such that $$ lcm \, (n, n + 1, n + 2, ... , n + k) > lcm \, (n + 1, n + 2, n + 3,... , n + k + 1).$$

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.

Two positive integers $m, n$ satisfy the two equations $m^2 + n^2 = 3789$ and $gcd (m, n) + lcm (m, n) = 633$. Compute $m + n$.

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 $a + b + c + d = 1000$?

Determine if there exist five consecutive positive integers such that their LCM is a perfect square.

Let $a < b < c < d < e$ be positive integers. Prove that $$\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1$$ where $[x, y]$ is the least common multiple of $x$ and $y$ (e.g., $[6, 10] = 30$). When does equality hold?

Let $n \ge 2$ be a natural number. For an $n$-element subset $F$ of $\{1, . . . , 2n\}$ we define $m(F)$ as the minimum of all $lcm \,\, (x, y)$ , where $x$ and $y$ are two distinct elements of $F$. Find the maximum value of $m(F)$.

Find the least possible value for the fraction $$\frac{lcm(a,b)+lcm(b,c)+lcm(c,a)}{gcd(a,b)+gcd(b,c)+gcd(c,a)}$$ over all distinct positive integers $a, b, c$. By $lcm(x, y)$ we mean the least common multiple of $x, y$ and by $gcd(x, y)$ we mean the greatest common divisor of $x, y$.

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 $b$.

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) $n = 2010$.

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.

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 $a_1 >\left[\frac{2n}{3}\right]$.

Find the number of pairs of positive integers $(m; n)$, with $m \le n$, such that the ‘least common multiple’ (LCM) of $m$ and $n$ equals $600$.

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]

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 $k$.

Determine all natural integers $n$ for which there is no triplet $(a, b, c)$ of natural numbers such that: $$n = \frac{a \cdot \,\,lcm(b, c) + b \cdot lcm \,\,(c, a) + c \cdot lcm \,\, (a, b)}{lcm \,\,(a, b, c)}$$

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 $i = 0, 1, 2$.

$S$ 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.