This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 121

PEN A Problems, 48
Tags: Divisibility Theory , pen
Let $n$ be a positive integer. Prove that \[\frac{1}{3}+\cdots+\frac{1}{2n+1}\] is not an integer.

PEN A Problems, 98
Tags: Divisibility Theory
Let $n$ be a positive integer with $k\ge22$ divisors $1=d_{1}< d_{2}< \cdots < d_{k}=n$, all different. Determine all $n$ such that \[{d_{7}}^{2}+{d_{10}}^{2}= \left( \frac{n}{d_{22}}\right)^{2}.\]

PEN A Problems, 23
Tags: quadratics , Gauss , modular arithmetic , number theory , relatively prime , Divisibility Theory , pen
(Wolstenholme's Theorem) Prove that if \[1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\] is expressed as a fraction, where $p \ge 5$ is a prime, then $p^{2}$ divides the numerator.

PEN A Problems, 36
Tags: floor function , Divisibility Theory
Let $n$ and $q$ be integers with $n \ge 5$, $2 \le q \le n$. Prove that $q-1$ divides $\left\lfloor \frac{(n-1)!}{q}\right\rfloor $.

PEN A Problems, 76
Tags: Divisibility Theory
Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[(a-1)(b-1)(c-1)\hspace{0.2in}\text{is a divisor of}\hspace{0.2in}abc-1.\]

PEN A Problems, 22
Tags: algebra , polynomial , binomial theorem , Divisibility Theory
Prove that the number \[\sum_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}\] is not divisible by $5$ for any integer $n\geq 0$.

PEN A Problems, 111
Tags: Divisibility Theory
Find all natural numbers $n$ such that the number $n(n+1)(n+2)(n+3)$ has exactly three different prime divisors.

PEN A Problems, 100
Tags: calculus , integration , modular arithmetic , Divisibility Theory
Find all positive integers $n$ such that $n$ has exactly $6$ positive divisors $1<d_{1}<d_{2}<d_{3}<d_{4}<n$ and $1+n=5(d_{1}+d_{2}+d_{3}+d_{4})$.

PEN A Problems, 113
Tags: number theory , relatively prime , Divisibility Theory
Find all triples $(l, m, n)$ of distinct positive integers satisfying \[{\gcd(l, m)}^{2}= l+m, \;{\gcd(m, n)}^{2}= m+n, \; \text{and}\;\;{\gcd(n, l)}^{2}= n+l.\]

PEN A Problems, 29
Tags: Divisibility Theory
For which positive integers $k$, is it true that there are infinitely many pairs of positive integers $(m, n)$ such that \[\frac{(m+n-k)!}{m! \; n!}\] is an integer?

PEN A Problems, 83
Tags: floor function , Divisibility Theory
Find all $n \in \mathbb{N}$ such that $ \lfloor \sqrt{n}\rfloor$ divides $n$.

PEN A Problems, 18
Tags: quadratics , algebra , difference of squares , special factorizations , Divisibility Theory
Let $m$ and $n$ be natural numbers and let $mn+1$ be divisible by $24$. Show that $m+n$ is divisible by $24$.

PEN A Problems, 90
Tags: Divisibility Theory
Determine all pairs $(x, y)$ of positive integers with $y \vert x^2 +1$ and $x^2 \vert y^3 +1$.

PEN A Problems, 30
Tags: Divisibility Theory
Show that if $n \ge 6$ is composite, then $n$ divides $(n-1)!$.

2017 Pan-African Shortlist, N1
Tags: number theory , greatest common divisor , floor function , function , algebra , Divisibility Theory
Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.

PEN A Problems, 19
Tags: induction , modular arithmetic , LaTeX , Divisibility Theory , number theory
Let $f(x)=x^3 +17$. Prove that for each natural number $n \ge 2$, there is a natural number $x$ for which $f(x)$ is divisible by $3^n$ but not $3^{n+1}$.

PEN A Problems, 27
Tags: modular arithmetic , binomial coefficients , Divisibility Theory
Show that the coefficients of a binomial expansion $(a+b)^n$ where $n$ is a positive integer, are all odd, if and only if $n$ is of the form $2^{k}-1$ for some positive integer $k$.

PEN A Problems, 117
Tags: Divisibility Theory , pen
Find the smallest positive integer $n$ such that \[2^{1989}\; \vert \; m^{n}-1\] for all odd positive integers $m>1$.

PEN A Problems, 7
Tags: quadratics , LaTeX , modular arithmetic , Divisibility Theory
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.

PEN A Problems, 2
Tags: arithmetic sequence , Diophantine equation , Divisibility Theory , Pell s Equation
Find infinitely many triples $(a, b, c)$ of positive integers such that $a$, $b$, $c$ are in arithmetic progression and such that $ab+1$, $bc+1$, and $ca+1$ are perfect squares.

PEN A Problems, 10
Tags: Euler , number theory , Divisibility Theory
Let $n$ be a positive integer with $n \ge 3$. Show that \[n^{n^{n^{n}}}-n^{n^{n}}\] is divisible by $1989$.

PEN A Problems, 34
Tags: Divisibility Theory
Let $p_{1}, p_{2}, \cdots, p_{n}$ be distinct primes greater than $3$. Show that \[2^{p_{1}p_{2}\cdots p_{n}}+1\] has at least $4^{n}$ divisors.

PEN A Problems, 55
Tags: Divisibility Theory
Show that for every natural number $n$ the product \[\left( 4-\frac{2}{1}\right) \left( 4-\frac{2}{2}\right) \left( 4-\frac{2}{3}\right) \cdots \left( 4-\frac{2}{n}\right)\] is an integer.

PEN A Problems, 40
Tags: number theory , greatest common divisor , Divisibility Theory
Determine the greatest common divisor of the elements of the set \[\{n^{13}-n \; \vert \; n \in \mathbb{Z}\}.\]

PEN A Problems, 91
Tags: Divisibility Theory
Determine all pairs $(a, b)$ of positive integers such that $ab^2+b+7$ divides $a^2 b+a+b$.