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, 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, 41
Tags: modular arithmetic , divisibility theory
Show that there are infinitely many composite numbers $n$ such that $3^{n-1}-2^{n-1}$ is divisible by $n$.

PEN A Problems, 116
Tags: divisibility theory
What is the smallest positive integer that consists base 10 of each of the ten digits, each used exactly once, and is divisible by each of the digits $2$ through $9$?

PEN A Problems, 102
Tags: divisibility theory
Determine all three-digit numbers $N$ having the property that $N$ is divisible by $11,$ and $\frac{N}{11}$ is equal to the sum of the squares of the digits of $N.$

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

2017 Pan-African Shortlist, N1
Tags: function , algebra , number theory , greatest common divisor , floor function , 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, 11
Tags: pigeonhole principle , linear algebra , modular arithmetic , matrix , divisibility theory
Let $a, b, c, d$ be integers. Show that the product \[(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)\] is divisible by $12$.

PEN A Problems, 33
Tags: divisibility theory
Let $a,b,x\in \mathbb{N}$ with $b>1$ and such that $b^{n}-1$ divides $a$. Show that in base $b$, the number $a$ has at least $n$ non-zero digits.

PEN A Problems, 22
Tags: polynomial , algebra , 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, 17
Tags: modular arithmetic , number theory , quadratic , relatively prime , divisibility theory
Let $m$ and $n$ be natural numbers such that \[A=\frac{(m+3)^{n}+1}{3m}\] is an integer. Prove that $A$ is odd.

PEN A Problems, 4
Tags: algebra , polynomial , search , vieta , symmetry , quadratic , divisibility theory
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.

PEN A Problems, 70
Tags: induction , divisibility theory , binomial coefficient
Suppose that $m=nq$, where $n$ and $q$ are positive integers. Prove that the sum of binomial coefficients \[\sum_{k=0}^{n-1}{ \gcd(n, k)q \choose \gcd(n, k)}\] is divisible by $m$.

PEN A Problems, 50
Tags: divisibility theory
Show that every integer $k>1$ has a multiple less than $k^4$ whose decimal expansion has at most four distinct digits.

PEN A Problems, 47
Tags: floor function , number theory , factorial , prime number , divisibility theory
Let $n$ be a positive integer with $n>1$. Prove that \[\frac{1}{2}+\cdots+\frac{1}{n}\] is not an integer.

PEN A Problems, 96
Tags: number theory , modular arithmetic , integration , relatively prime , divisibility theory
Find all positive integers $n$ that have exactly $16$ positive integral divisors $d_{1},d_{2} \cdots, d_{16}$ such that $1=d_{1}<d_{2}<\cdots<d_{16}=n$, $d_6=18$, and $d_{9}-d_{8}=17$.

PEN A Problems, 38
Tags: floor function , divisibility theory
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 $1<a<b$.

PEN A Problems, 108
Tags: arithmetic sequence , divisibility theory
For each integer $n>1$, let $p(n)$ denote the largest prime factor of $n$. Determine all triples $(x, y, z)$ of distinct positive integers satisfying [list] [*] $x, y, z$ are in arithmetic progression, [*] $p(xyz) \le 3$. [/list]

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, 114
Tags: greatest common divisor , modular arithmetic , number theory , divisibility theory
What is the greatest common divisor of the set of numbers \[\{{16}^{n}+10n-1 \; \vert \; n=1,2,\cdots \}?\]

PEN A Problems, 86
Tags: inequalities , induction , divisibility theory
Find all positive integers $(x, n)$ such that $x^{n}+2^{n}+1$ divides $x^{n+1}+2^{n+1}+1$.

PEN A Problems, 23
Tags: number theory , quadratic , gauss , modular arithmetic , relatively prime , divisibility theory
(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.