Olympiad problems

2004 AMC 10, 4

Tags: least common multiple

A standard six-sided die is rolled, and $ P$ is the product of the five numbers that are visible. What is the largest number that is certain to divide $ P$? $ \textbf{(A)}\ 6\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 24\qquad \textbf{(D)}\ 144\qquad \textbf{(E)}\ 720$

1993 AIME Problems, 6

Tags: number theory , least common multiple

What is the smallest positive integer than can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?

1997 Canada National Olympiad, 1

Tags: least common multiple , number theory

Determine the number of pairs of positive integers $x,y$ such that $x\le y$, $\gcd (x,y)=5!$ and $\text{lcm}(x,y)=50!$.

2014 India National Olympiad, 3

Tags: greatest common divisor , least common multiple , inequalities , number theory

Let $a,b$ be natural numbers with $ab>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisble by $a+b$. Prove that the quotient is at most $\frac{a+b}{4}$. When is this quotient exactly equal to $\frac{a+b}{4}$

2005 AMC 8, 20

Tags: number theory , least common multiple , modular arithmetic

Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take? $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 24 $

2001 Saint Petersburg Mathematical Olympiad, 10.6

Tags: inequalities , gcd , number theory , algebra , greatest common divisor , least common multiple

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]

2021 Baltic Way, 20

Tags: divisibility , least common multiple , number theory

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 $$a_1a_2\ldots a_n \mid (n+1)(n+2)\ldots (2n-1)(2n).$$

2023 All-Russian Olympiad Regional Round, 9.6

Tags: number theory , least common multiple

Does there exist a positive integer $m$, such that if $S_n$ denotes the lcm of $1,2, \ldots, n$, then $S_{m+1}=4S_m$?

2017 CCA Math Bonanza, L2.4

Tags: number theory , least common multiple

Define $f\left(n\right)=\textrm{LCM}\left(1,2,\ldots,n\right)$. Determine the smallest positive integer $a$ such that $f\left(a\right)=f\left(a+2\right)$. [i]2017 CCA Math Bonanza Lightning Round #2.4[/i]

1990 Romania Team Selection Test, 3

Tags: number theory , least common multiple , combinatorics

Prove that for any positive integer $n$, the least common multiple of the numbers $1,2,\ldots,n$ and the least common multiple of the numbers: \[\binom{n}{1},\binom{n}{2},\ldots,\binom{n}{n}\] are equal if and only if $n+1$ is a prime number. [i]Laurentiu Panaitopol[/i]

2006 AIME Problems, 14

Tags: number theory , least common multiple , relatively prime

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.

1987 AMC 8, 9

Tags: number theory , least common multiple , relatively prime , prime number

When finding the sum $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}$, the least common denominator used is $\text{(A)}\ 120 \qquad \text{(B)}\ 210 \qquad \text{(C)}\ 420 \qquad \text{(D)}\ 840 \qquad \text{(E)}\ 5040$

2014 Kyiv Mathematical Festival, 4a

Tags: least common multiple , modular arithmetic , number theory

a) Prove that for every positive integer $y$ the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=x(x+1)$ holds for infinitely many positive integers $x.$ b) Prove that there exists positive integer $y$ such that the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=y(y+1)$ holds for at least 2014 positive integers $x.$

2010 Portugal MO, 1

Tags: least common multiple , number theory

Giraldo wrote five distinct natural numbers on the vertices of a pentagon. And next he wrote on each side of the pentagon the least common multiple of the numbers written of the two vertices who were on that side and noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides?

2006 Iran MO (3rd Round), 1

Tags: modular arithmetic , abstract algebra , least common multiple , function , group theory , number theory

$n$ is a natural number. $d$ is the least natural number that for each $a$ that $gcd(a,n)=1$ we know $a^{d}\equiv1\pmod{n}$. Prove that there exist a natural number that $\mbox{ord}_{n}b=d$

1989 AMC 8, 22

Tags: number theory , least common multiple

The letters $\text{A}$, $\text{J}$, $\text{H}$, $\text{S}$, $\text{M}$, $\text{E}$ and the digits $1$, $9$, $8$, $9$ are "cycled" separately as follows and put together in a numbered list: \[\begin{tabular}[t]{lccc} & & AJHSME & 1989 \\ & & & \\ 1. & & JHSMEA & 9891 \\ 2. & & HSMEAJ & 8919 \\ 3. & & SMEAJH & 9198 \\ & & ........ & \end{tabular}\] What is the number of the line on which $\text{AJHSME 1989}$ will appear for the first time? $\text{(A)}\ 6 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 24$

1951 AMC 12/AHSME, 37

Tags: number theory , least common multiple

A number which when divided by $ 10$ leaves a remainder of $ 9$, when divided by $ 9$ leaves a remainder of $ 8$, by $ 8$ leaves a remainder of $ 7$, etc., down to where, when divided by $ 2$, it leaves a remainder of $ 1$, is: $ \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 419 \qquad\textbf{(C)}\ 1259 \qquad\textbf{(D)}\ 2519 \qquad\textbf{(E)}\ \text{none of these answers}$

1990 AIME Problems, 10

Tags: trigonometry , number theory , least common multiple , greatest common divisor , modular arithmetic , complex number , relatively prime

The sets $A = \{z : z^{18} = 1\}$ and $B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A \ \text{and} \ w \in B\}$ is also a set of complex roots of unity. How many distinct elements are in $C$?

2020 Latvia Baltic Way TST, 16

Tags: number theory , least common multiple , greatest common divisor

Given sequence $\{a_n\}$ satisfying: $$ a_{n+1} = \frac{ lcm(a_n,a_{n-1})}{\gcd(a_n, a_{n-1})} $$ It is given that $a_{209} =209$ and $a_{361} = 361$. Find all possible values of $a_{2020}$.

2022 AMC 10, 19

Tags: least common multiple

Define $L_n$ as the least common multiple of all the integers from $1$ to $n$ inclusive. There is a unique integer $h$ such that $\frac{1}{1}+\frac{1}{2}+\frac{1}{3} \ldots +\frac{1}{17}=\frac{h}{L_{17}}$. What is the remainder when $h$ is divided by $17?$ $\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$

PEN O Problems, 7

Tags: induction , number theory , least common multiple

Show that for each $n \ge 2$, there is a set $S$ of $n$ integers such that $(a-b)^2$ divides $ab$ for every distinct $a, b\in S$.

2004 Dutch Mathematical Olympiad, 1

Tags: number theory , least common multiple

Determine the number of pairs of positive integers $(a, b)$, with $a \le b$, for which lcm $(a, b) = 2004$. lcm ($a, b$) means the least common multiple of $a$ and $b$. Example: lcm $(18, 24) = 72$.

1999 Putnam, 6

Tags: least common multiple , relatively prime , prime number , putnam number theory , number theory

Let $S$ be a finite set of integers, each greater than $1$. Suppose that for each integer $n$ there is some $s\in S$ such that $\gcd(s,n)=1$ or $\gcd(s,n)=s$. Show that there exist $s,t\in S$ such that $\gcd(s,t)$ is prime.

2009 Harvard-MIT Mathematics Tournament, 4

Tags: number theory , least common multiple , greatest common divisor , algebra , polynomial , quadratic

Suppose $a$, $b$ and $c$ are integers such that the greatest common divisor of $x^2+ax+b$ and $x^2+bx+c$ is $x+1$ (in the set of polynomials in $x$ with integer coefficients), and the least common multiple of $x^2+ax+b$ and $x^2+bx+c$ $x^3-4x^2+x+6$. Find $a+b+c$.

2011 AMC 10, 23

Tags: modular arithmetic , euler , function , number theory , least common multiple , search

What is the hundreds digit of $2011^{2011}$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9 $