Found problems: 583
Compute gcd $\left( \frac{135^{90}-45^{90}}{90^2} , 90^2 \right)$
Compute \[\sum_{k=1}^{420} \gcd(k,420).\]
[i]2016 CCA Math Bonanza Individual #14[/i]
Find all pairs $(n, m)$ of positive integers such that $gcd ((n + 1)^m - n, (n + 1)^{m+3} - n) > 1$.
Suppose that both $x^{3}-x$ and $x^{4}-x$ are integers for some real number $x$. Show that $x$ is an integer.
Do there exist a positive integer $k$ and a non-constant sequence $a_1, a_2, a_3, ...$ of positive integers such that $a_n = gcd(a_{n+k}, a_{n+k+1})$ for all positive integers $n$?
A finite set $S$ of positive integers has the property that, for each $s \in S,$ and each positive integer divisor $d$ of $s$, there exists a unique element $t \in S$ satisfying $\text{gcd}(s, t) = d$. (The elements $s$ and $t$ could be equal.)
Given this information, find all possible values for the number of elements of $S$.
Is it possible for Pingu to choose $2025$ positive integers $a_1, ..., a_{2025}$ such that:
1. The sequence $a_i$ is increasing;
2. $\gcd(a_1,a_2)>\gcd(a_2,a_3)>...>\gcd(a_{2024},a_{2025})>\gcd(a_{2025},a_1)>1$?
[i](Proposed by Tan Rui Xuen and Ivan Chan Guan Yu)[/i]
Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is:
\[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\]
Find the least possible value of $n$.
Find all the pairs positive integers $(x, y)$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{[x, y]}+\frac{1}{(x, y)}=\frac{1}{2}$ ,
where $(x, y)$ is the greatest common divisor of $x, y$ and $[x, y]$ is the least common multiple of $x, y$.
Let $d$ be a positive divisor of a positive integer $m$ and $(a_l), (b_l)$ two arithmetic sequences of positive integers. It is given that $gcd(a_i, b_j) = 1$ and $gcd(a_k, b_n) = m$ for some positive integers $i,j,k,$ and $n$. Prove that there exist positive integers $t$ and $s$ such that $gcd(a_t, b_s) = d$.
Prove that for all natural $n$ there exists $a,b,c$ such that $n=\gcd (a,b)(c^2-ab)+\gcd (b,c)(a^2-bc)+\gcd (c,a)(b^2-ca)$.
For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$.
Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.
Given three positive integers $a,b,$ and $c$. Their greatest common divisor is $D$; their least common multiple is $m$. Then, which two of the following statements are true?
$ \text{(1)}\ \text{the product MD cannot be less than abc} \qquad$
$\text{(2)}\ \text{the product MD cannot be greater than abc}\qquad$
$\text{(3)}\ \text{MD equals abc if and only if a,b,c are each prime}\qquad$
$\text{(4)}\ \text{MD equals abc if and only if a,b,c are each relatively prime in pairs}$ $\text{ (This means: no two have a common factor greater than 1.)}$
$ \textbf{(A)}\ 1,2 \qquad\textbf{(B)}\ 1,3\qquad\textbf{(C)}\ 1,4\qquad\textbf{(D)}\ 2,3\qquad\textbf{(E)}\ 2,4 $
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 $\frac{a+b}{4}$
Two people play a game as follows: At the beginning both of them have one point and in every move, one of them can double it's points, or when the other have more point than him, subtract to him his points. Can the two competitors have 2009 and 2002 points respectively? What about 2009 and 2003? Generally which couples of points can they have?
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.
For natural numbers $x$ and $y$, let $(x,y)$ denote the greatest common divisor of $x$ and $y$. How many pairs of natural numbers $x$ and $y$ with $x \le y$ satisfy the equation $xy = x + y + (x, y)$?
Suppose $a$, $b$, and $c$ are positive integers such that \[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\] Which of the following statements are necessarily true?
I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$.
II. If $\gcd(c,210)=1$, then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both.
III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$.
$\textbf{(A)}~\text{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II and III only}$
Prove that if $N{}$ is a large enough positive integer, then for any permutation $\pi_1,\ldots,\pi_N$ of $1,\ldots, N$ at least $11\%$ of the pairs $(i,j)$ of indices from $1{}$ to $N{}$ satisfy $\gcd(i,j)=1=\gcd(\pi_i,\pi_j).$
[i]Proposed by Vlad Spătaru[/i]
Let $X$ be the set of natural numbers whose all digits in the decimal representation are different. For $n \in N$, denote by $A_n$ the set of numbers whose digits are a permutation of the digits of $n$, and $d_n$ be the greatest common divisor of the numbers in $A_n$. (For example, $A_{1120} =\{112,121,...,2101,2110\}$, so $d_{1120} = 1$.)
Find the maximum possible value of $d_n$.
Determine all pairs $(x, y)$ of positive integers such that for $d = gcd(x, y)$ the equation $$xyd = x + y + d^2$$
holds.
[i](Walther Janous)[/i]
a) If $K$ is a finite extension of the field $F$ and $K=F(\alpha,\beta)$ show that $[K: F]\leq [F(\alpha): F][F(\beta): F]$
b) If $gcd([F(\alpha): F],[F(\beta): F])=1$ then does the above inequality always become equality?
c) By giving an example show that if $gcd([F(\alpha): F],[F(\beta): F])\neq 1$ then equality might happen.
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2016$ good partitions.
PS. [url=https://artofproblemsolving.com/community/c6h1268855p6622233]2015 ISL C3 [/url] has 2015 instead of 2016
Let $19$ integer numbers are given. Let Hamza writes on the paper the greatest common divisor for each pair of numbers. It occurs that the difference between the biggest and smallest numbers written on the paper is less than $180$. Prove that not all numbers on the paper are different.
Let $a, b$ and $c$ be natural numbers. Determine the smallest value that the following expression can take:
$$\frac{a}{gcd\,\,(a + b, a - c)}
+
\frac{b}{gcd\,\,(b + c, b - a)}
+
\frac{c}{gcd\,\,(c + a, c - b)}.$$
.
Remark: $gcd \,\, (6, 0) = 6$ and $gcd\,\,(3, -6) = 3$.