Found problems: 583
Determine the number of ordered pairs of positive integers $(m,n)$ with $1\leq m\leq 100$ and $1\leq n\leq 100$ such that
\[
\gcd(m+1,n+1) = 10\gcd(m,n).
\]
The measure of a given angle is $\frac{180^{\circ}}{n}$ where $n$ is a positive integer not divisible by $3$. Prove that the angle can be trisected by Euclidean means (straightedge and compasses).
Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$.
[i]Proposed by Morteza Saghafian, Iran[/i]
Determine all finite nonempty sets $S$ of positive integers satisfying
\[ {i+j\over (i,j)}\qquad\mbox{is an element of S for all i,j in S}, \]
where $(i,j)$ is the greatest common divisor of $i$ and $j$.
Let $d$ denote the greatest common divisor of the natural numbers $a$ and $b$, and let $d'$ denote the greatest common divisor of the natural numbers $a'$ and $b'$. Prove that $dd'$ is the greatest common divisor of the four numbers $$ aa' , \ \ ab' , \ \ ba' , \ \ bb' .$$
Jamar bought some pencils costing more than a penny each at the school bookstore and paid $\$1.43$. Sharona bought some of the same pencils and paid $\$1.87$. How many more pencils did Sharona buy than Jamar?
$\textbf{(A)}\hspace{.05in}2 \qquad \textbf{(B)}\hspace{.05in}3 \qquad \textbf{(C)}\hspace{.05in}4 \qquad \textbf{(D)}\hspace{.05in}5 \qquad \textbf{(E)}\hspace{.05in}6 $
For every integers $ a,b,c$ whose greatest common divisor is $n$, if
\[ \begin{array}{l} {x \plus{} 2y \plus{} 3z \equal{} a} \\
{2x \plus{} y \minus{} 2z \equal{} b} \\
{3x \plus{} y \plus{} 5z \equal{} c} \end{array}
\]
has a solution in integers, what is the smallest possible value of positive number $ n$?
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 14 \qquad\textbf{(C)}\ 28 \qquad\textbf{(D)}\ 56 \qquad\textbf{(E)}\ \text{None}$
What is the units digit of $ 13^{2003}$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 9$
Find the number of 8-digit base-6 positive integers $(a_1a_2a_3a_4a_5a_6a_7a_8)_6$ (with leading zeros permitted) such that $(a_1a_2\ldots a_8)_6\mid(a_{i+1}a_{i+2}\ldots a_{i+8})_6$ for $i=1,2,\ldots,7$, where indices are taken modulo $8$ (so $a_9=a_1$, $a_{10}=a_2$, and so on).
[i]Victor Wang[/i]
Suppose N is a $6$-digit number having base-$10$ representation $\underline{a}\text{ }\underline{b}\text{ }\underline{c}\text{ }\underline{d}\text{ }\underline{e}\text{ }\underline{f}$. If $N$ is $6/7$ of the number having base-$10$ representation $\underline{d}\text{ }\underline{e}\text{ }\underline{f}\text{ }\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, find $N$.
Let $n$ be a positive integer. Determine the maximum value of $gcd(a, b) + gcd(b, c) + gcd(c, a)$ for positive integers $a, b, c$ such that $a + b + c = 5n$.
Let $d_1$ and $d_2$ be divisors of a positive integer $n$. Suppose that the greatest common divisor of $d_1$ and $n/d_2$ and the greatest common divisor of $d_2$ and $n/d_1$ are equal. Show that $d_1 = d_2$.
Does there exist positive integers $a_{1}<a_{2}<\cdots<a_{100}$ such that for $2 \le k \le 100$, the greatest common divisor of $a_{k-1}$ and $a_{k}$ is greater than the greatest common divisor of $a_{k}$ and $a_{k+1}$?
Prove that the equation $x^3+y^3+z^3=t^4$ has infinitely many solutions in positive integers such that $\gcd(x,y,z,t)=1$.
[i]Mihai Pitticari & Sorin Rǎdulescu[/i]
Let $ a > b > 1$ be relatively prime positive integers. Define the weight of an integer $ c$, denoted by $ w(c)$ to be the minimal possible value of $ |x| \plus{} |y|$ taken over all pairs of integers $ x$ and $ y$ such that \[ax \plus{} by \equal{} c.\] An integer $ c$ is called a [i]local champion [/i]if $ w(c) \geq w(c \pm a)$ and $ w(c) \geq w(c \pm b)$.
Find all local champions and determine their number.
[i]Proposed by Zoran Sunic, USA[/i]
The sequence of integers an is given by $a_0 = 0, a_n = p(a_n-1)$, where $p(x)$ is a polynomial whose coefficients are all positive integers. Show that for any two positive integers $m, k$ with greatest common divisor $d$, the greatest common divisor of $a_m$ and $a_k$ is $a_d$.
Find the number of ordered pairs $(m, n)$ such that $m$ and $n$ are positive integers in the set $\{1, 2, ..., 30\}$ and the greatest common divisor of $2^m + 1$ and $2^n - 1$ is not $1.$
If $a$ and $b$ are coprime positive integers and $n$ an integer, prove that the greatest common divisor of $a^2+b^2-nab$ and $a+b$ divides $n+2$.
Let $x,y,z$ be positive integers such that $\frac{x+1}{y}+\frac{y+1}{z}+\frac{z+1}{x}$ is an integer. Let $d$ be the greatest common divisor of $x,y$ and $z$. Prove that $d\le \sqrt[3]{xy+yz+zx}$.
The difference between the product and the sum of two different integers is equal to the sum of their GCD (greatest common divisor) and LCM (least common multiple). Findall these pairs of numbers. Justify your answer.
Find all ordered positive integer pairs of $(m,n)$ such that $2^n-1$ divides $2^m+1$.
Let $T$ be a non-empty finite subset of positive integers $\ge 1$. A subset $S$ of $T$ is called [b]good [/b] if for every integer $t\in T$ there exists an $s$ in $S$ such that $gcd(t,s) >1$. Let
\[A={(X,Y)\mid X\subseteq T,Y\subseteq T,gcd(x,y)=1 \text{for all} x\in X, y\in Y}\]
Prove that :
$a)$ If $X_0$ is not [b]good[/b] then the number of pairs $(X_0,Y)$ in $A$ is [b]even[/b].
$b)$ the number of good subsets of $T$ is [b]odd[/b].
Consider $2$ positive integers $a,b$ such that $a+2b=2020$.
(a) Determine the largest possible value of the greatest common divisor of $a$ and $b$.
(b) Determine the smallest possible value of the least common multiple of $a$ and $b$.
Let $a$ be the smallest and $A$ the largest of $n$ distinct positive integers. Prove that the least common multiple of these numbers is greater than or equal to $n a$ and that the greatest common divisor is less than or equal to $\frac{A}{n}$
For any positive integer $ n>1$, let $ P\left(n\right)$ denote the largest prime divisor of $ n$. Prove that there exist infinitely many positive integers $ n$ for which
\[ P\left(n\right)<P\left(n\plus{}1\right)<P\left(n\plus{}2\right).\]