Found problems: 121
Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.
Find all pairs of positive integers $m, n \ge 3$ for which there exist infinitely many positive integers $a$ such that \[\frac{a^{m}+a-1}{a^{n}+a^{2}-1}\] is itself an integer.
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 \[\text{lcm}(a_{1}, \cdots, a_{r})=\prod_{i=1}^{r}\prod_{s\in S_{i}}\gcd(s)^{\left((-1)^{i}\right)}.\]
Find an integer $n$, where $100 \leq n \leq 1997$, such that \[\frac{2^{n}+2}{n}\] is also an integer.
Determine the greatest common divisor of the elements of the set \[\{n^{13}-n \; \vert \; n \in \mathbb{Z}\}.\]
Prove that there exist infinitely many pairs $(a, b)$ of relatively prime positive integers such that \[\frac{a^{2}-5}{b}\;\; \text{and}\;\; \frac{b^{2}-5}{a}\] are both positive integers.
Let $m$ and $n$ be natural numbers and let $mn+1$ be divisible by $24$. Show that $m+n$ is divisible by $24$.
Find all natural numbers $n$ such that the number $n(n+1)(n+2)(n+3)$ has exactly three different prime divisors.
If $n$ is a natural number, prove that the number $(n+1)(n+2)\cdots(n+10)$ is not a perfect square.
If $a, b, c$ are positive integers such that \[0 < a^{2}+b^{2}-abc \le c,\] show that $a^{2}+b^{2}-abc$ is a perfect square.
Let $n$ be a positive integer such that $2+2\sqrt{28n^2 +1}$ is an integer. Show that $2+2\sqrt{28n^2 +1}$ is the square of an integer.
A wobbly number is a positive integer whose $digits$ in base $10$ are alternatively non-zero and zero the units digit being non-zero. Determine all positive integers which do not divide any wobbly number.
Find four positive integers, each not exceeding $70000$ and each having more than $100$ divisors.
Let $p$ be a prime with $p>5$, and let $S=\{p-n^2 \vert n \in \mathbb{N}, {n}^{2}<p \}$. Prove that $S$ contains two elements $a$ and $b$ such that $a \vert b$ and $1<a<b$.
Let $n \ge 2$ be a positive integer, with divisors \[1=d_{1}< d_{2}< \cdots < d_{k}=n \;.\] Prove that \[d_{1}d_{2}+d_{2}d_{3}+\cdots+d_{k-1}d_{k}\] is always less than $n^{2}$, and determine when it divides $n^{2}$.
Let $ a$ and $ b$ be natural numbers such that
\[ \frac{a}{b}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots-\frac{1}{1318}+\frac{1}{1319}. \]
Prove that $ a$ is divisible by $ 1979$.
Let n be a positive integer. Show that the product of $ n$ consecutive positive integers is divisible by $ n!$
Let $a$ and $b$ be integers. Show that $a$ and $b$ have the same parity if and only if there exist integers $c$ and $d$ such that $a^2 +b^2 +c^2 +1 = d^2$.
A natural number $n$ is said to have the property $P$, if whenever $n$ divides $a^{n}-1$ for some integer $a$, $n^2$ also necessarily divides $a^{n}-1$. [list=a] [*] Show that every prime number $n$ has the property $P$. [*] Show that there are infinitely many composite numbers $n$ that possess the property $P$. [/list]
Let $n$ be a positive integer. Prove that \[\frac{1}{3}+\cdots+\frac{1}{2n+1}\] is not an integer.
Let $p>3$ is a prime number and $k=\lfloor\frac{2p}{3}\rfloor$. Prove that \[{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}\] is divisible by $p^{2}$.
Find all $n \in \mathbb{N}$ such that $ \lfloor \sqrt{n}\rfloor$ divides $n$.
Show that there are infinitely many composite numbers $n$ such that $3^{n-1}-2^{n-1}$ is divisible by $n$.
Does there exist a $4$-digit integer (in decimal form) such that no replacement of three of its digits by any other three gives a multiple of $1992$?
Let $a$ and $b$ be positive integers. When $a^{2}+b^{2}$ is divided by $a+b,$ the quotient is $q$ and the remainder is $r.$ Find all pairs $(a,b)$ such that $q^{2}+r=1977$.