Found problems: 283
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 $600$.
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.
Let $ n$ be a positive integer, let $ A$ be a subset of $ \{1, 2, \cdots, n\}$, satisfying for any two numbers $ x, y\in A$, the least common multiple of $ x$, $ y$ not more than $ n$. Show that $ |A|\leq 1.9\sqrt {n} \plus{} 5$.
Find all positive integers $k$ for which the equation: $$ \text{lcm}(m,n)-\text{gcd}(m,n)=k(m-n)$$ has no solution in integers positive $(m,n)$ with $m\neq n$.
How many ordered pairs $(a, b)$ of positive integers satisfy the equation
$$a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),$$
where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ denotes their least common multiple?
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8$
Find the sum of the greatest common factor and the least common multiple of $12$ and $18$.
Assume that $a_1, a_2, a_3$ are three given positive integers consider the following sequence:
$a_{n+1}=\text{lcm}[a_n, a_{n-1}]-\text{lcm}[a_{n-1}, a_{n-2}]$ for $n\ge 3$
Prove that there exist a positive integer $k$ such that $k\le a_3+4$ and $a_k\le 0$.
($[a, b]$ means the least positive integer such that$ a\mid[a,b], b\mid[a, b]$ also because $\text{lcm}[a, b]$ takes only nonzero integers this sequence is defined until we find a zero number in the sequence)
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)
Find all natural numbers $n$ greater than $2$ such that there exist $n$ natural numbers $a_{1},a_{2},\ldots,a_{n}$ such that they are not all equal, and the sequence $a_{1}a_{2},a_{2}a_{3},\ldots,a_{n}a_{1}$ forms an arithmetic progression with nonzero common difference.
Find all posible pairs of positive integers $x,y$ such that $$\text{lcm}(x,y+3001)=\text{lcm}(y,x+3001)$$
If $ n$ is any whole number, $ n^2(n^2 \minus{} 1)$ is always divisible by
$ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 24 \qquad\textbf{(C)}\ \text{any multiple of }12 \qquad\textbf{(D)}\ 12 \minus{} n \qquad\textbf{(E)}\ 12\text{ and }24$
Three planets revolve about a star in coplanar circular orbits with the star at the center. All planets revolve in the same direction, each at a constant speed, and the periods of their orbits are 60, 84, and 140 years. The positions of the star and all three planets are currently collinear. They will next be collinear after $n$ years. Find $n$.
Alice and Bob are playing a guessing game. Bob is thinking of a number n of the form $2^a3^b$, where a and b are positive integers between $ 1$ and $2020$, inclusive. Each turn, Alice guess a number m, and Bob will tell her either $\gcd (m, n)$ or $lcm (m, n)$ (letting her know that he is saying that $gcd$ or $lcm$), as well as whether any of the respective powers match up in their prime factorization. In particular, if $m = n$, Bob will let Alice know this, and the game is over. Determine the smallest number $k$ so that Alice is always able to find $n$ within $k$ guesses, regardless of Bob’s number or choice of revealing either the $lcm$, or the $gcd$ .
Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.
An arbitrary prime $ p$ is given. If an integer sequence $ (n_1 , n_2 , \cdots , n_k )$ satisfying the conditions
- For all $ i\equal{} 1, 2, \cdots , k$, $ n_i \geq \frac{p\plus{}1}{2}$
- For all $ i\equal{} 1, 2, \cdots , k$, $ p^{n_i} \minus{} 1$ is divisible by $ n_{i\plus{}1}$, and $ \frac{p^{n_i} \minus{} 1}{n_{i\plus{}1}}$ is coprime to $ n_{i\plus{}1}$. Let $ n_{k\plus{}1} \equal{} n_1$.
exists not for $ k\equal{}1$, but exists for some $ k \geq 2$, then call the prime a good prime.
Prove that a prime is good iff it is not $ 2$.
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 $a + b + c + d = 1000$?
We call a sequence of $m$ consecutive integers a [i]friendly[/i] sequence if its first term is divisible by $1$, the second by $2$, ..., the $(m-1)^{th}$ by $m-1$, and in addition, the last term is divisible by $m^2$
Does a friendly sequence exist for (a) $m=20$ and (b) $m=11$?
A positive integer $n$ is said to be a [i]perfect power[/i] if $n=a^b$ for some integers $a,b$ with $b>1$.
$(\text{a})$ Find $2004$ perfect powers in arithmetic progression.
$(\text{b})$ Prove that perfect powers cannot form an infinite arithmetic progression.
Consider the sequence: $x_1=19,x_2=95,x_{n+2}=\text{lcm} (x_{n+1},x_n)+x_n$, for $n>1$, where $\text{lcm} (a,b)$ means the least common multiple of $a$ and $b$. Find the greatest common divisor of $x_{1995}$ and $x_{1996}$.
Show that ${2n \choose n} \; \vert \; \text{lcm}(1,2, \cdots, 2n)$ for all positive integers $n$.
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N} $ satisfying
\[ f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) \]
for all positive integer $m,n$.
Let $m$ and $n$ be positive integers such that $m - n = 189$ and such that the least common multiple of $m$ and $n$ is equal to $133866$. Find $m$ and $n$.
$\frac{2}{10}+\frac{4}{100}+\frac{6}{1000} =$
$\text{(A)}\ .012 \qquad \text{(B)}\ .0246 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .246 \qquad \text{(E)}\ 246$
Suppose that $S=\{a_{1}, \cdots, a_{r}\}$ is a set of positive integers, and let $S_{k}$ denote the set of subsets of $S$ with $k$ elements. Show that \[\text{lcm}(a_{1}, \cdots, a_{r})=\prod_{i=1}^{r}\prod_{s\in S_{i}}\gcd(s)^{\left((-1)^{i}\right)}.\]
Let $n\ge 2$ be an integer. Given numbers $a_1, a_2, \ldots, a_n \in \{1,2,3,\ldots,2n\}$ such that $\operatorname{lcm}(a_i,a_j)>2n$ for all $1\le i<j\le n$, prove that
$$a_1a_2\ldots a_n \mid (n+1)(n+2)\ldots (2n-1)(2n).$$