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.

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Found problems: 114

2019 Polish Junior MO First Round, 4
Tags: number theory , remainder
Positive integers $a, b, c$ have the property that: $\bullet$ $a$ gives remainder $2$ when divided by $b$, $\bullet$ $b$ gives remainder $2$ when divided by $c$, $\bullet$ $c$ gives remainder $4$ when divided by $a$. Prove that $c = 4$.

1981 Kurschak Competition, 3
Tags: remainder , number theory
For a positive integer $n$, $r(n)$ denote the sum of the remainders when $n$ is divided by $1, 2,..., n$ respectively. Prove that $r(k) = r(k -1)$ for infinitely many positive integers $k$.

2021 Saudi Arabia Training Tests, 32
Tags: sequence , remainder , number theory , algebra
Let $N$ be a positive integer. Consider the sequence $a_1, a_2, ..., a_N$ of positive integers, none of which is a multiple of $2^{N+1}$. For $n \ge N +1$, the number $a_n$ is defined as follows: choose $k$ to be the number among $1, 2, ..., n - 1$ for which the remainder obtained when $a_k$ is divided by $2^n$ is the smallest, and define $a_n = 2a_k$ (if there are more than one such $k$, choose the largest such $k$). Prove that there exist $M$ for which $a_n = a_M$ holds for every $n \ge M$.

2018 Hanoi Open Mathematics Competitions, 4
Tags: number theory , remainder , sum
A pyramid of non-negative integers is constructed as follows (a) The first row consists of only $0$, (b) The second row consists of $1$ and $1$, (c) The $n^{th}$ (for $n > 2$) is an array of $n$ integers among which the left most and right most elements are equal to $n - 1$ and the interior numbers are equal to the sum of two adjacent numbers from the $(n - 1)^{th}$ row (see Figure). Let $S_n$ be the sum of numbers in row $n^{th}$. Determine the remainder when dividing $S_{2018}$ by $2018$: A. $2$ B. $4$ C. $6$ D. $11$ E. $17$

2021 Saudi Arabia Training Tests, 30
Tags: number theory , remainder
For a positive integer $k$, denote by $f(k)$ the number of positive integer $m$ such that the remainder of $km$ modulo $2019^3$ is greater than $m$. Find the amount of different numbers among $f(1), f(2), ..., f(2019^3)$.

2019 Saudi Arabia JBMO TST, 3
Tags: remainder , number theory
Let $d$ be a positive divisor of the number $A = 1024^{1024}+5$ and suppose that $d$ can be expressed as $d = 2x^2+2xy+3y^2$ for some integers $x,y$. Which remainder we can have when divide $d$ by $20$ ?

2010 Junior Balkan Team Selection Tests - Romania, 3
Tags: number theory , factorial , remainder
Determine the integers $n, n \ge 2$, with the property that the numbers $1! , 2 ! , 3 ! , ..., (n- 1)!$ give different remainders when dividing by $n $.

2024 AMC 12/AHSME, 14
Tags: remainder
How many different remainders can result when the $100$th power of an integer is divided by $125$? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }5 \qquad \textbf{(D) }25 \qquad \textbf{(E) }125 \qquad $

2024 Romanian Master of Mathematics, 4
Tags: remainder , size , number theory
Fix integers $a$ and $b$ greater than $1$. For any positive integer $n$, let $r_n$ be the (non-negative) remainder that $b^n$ leaves upon division by $a^n$. Assume there exists a positive integer $N$ such that $r_n < \frac{2^n}{n}$ for all integers $n\geq N$. Prove that $a$ divides $b$. [i]Pouria Mahmoudkhan Shirazi, Iran[/i]

Kvant 2020, M2601
Tags: game , remainder , combinatorics
Gleb picked positive integers $N$ and $a$ ($a < N$). He wrote the number $a$ on a blackboard. Then each turn he did the following: he took the last number on the blackboard, divided the number $N$ by this last number with remainder and wrote the remainder onto the board. When he wrote the number $0$ onto the board, he stopped. Could he pick $N$ and $a$ such that the sum of the numbers on the blackboard would become greater than $100N$ ? Ivan Mitrofanov

2005 Taiwan TST Round 3, 3
Tags: modular arithmetic , divisibility , remainder , number theory
Given an integer ${n>1}$, denote by $P_{n}$ the product of all positive integers $x$ less than $n$ and such that $n$ divides ${x^2-1}$. For each ${n>1}$, find the remainder of $P_{n}$ on division by $n$. [i]Proposed by John Murray, Ireland[/i]

2004 IMO Shortlist, 6
Tags: remainder , number theory , divisibility , modular arithmetic
Given an integer ${n>1}$, denote by $P_{n}$ the product of all positive integers $x$ less than $n$ and such that $n$ divides ${x^2-1}$. For each ${n>1}$, find the remainder of $P_{n}$ on division by $n$. [i]Proposed by John Murray, Ireland[/i]

1994 North Macedonia National Olympiad, 1
Tags: number theory , sum , remainder
Let $ a_1, a_2, ..., a_ {1994} $ be integers such that $ a_1 + a_2 + ... + a_{1994} = 1994 ^{1994} $ . Determine the remainder of the division of $ a ^ 3_1 + a ^ 3_2 + ... + a ^ 3_{1994} $ with $6$.

1974 Bundeswettbewerb Mathematik, 4
Tags: number theory , divisor , odd , remainder
Peter and Paul gamble as follows. For each natural number, successively, they determine its largest odd divisor and compute its remainder when divided by $4$. If this remainder is $1$, then Peter gives Paul a coin; otherwise, Paul gives Peter a coin. After some time they stop playing and balance the accounts. Prove that Paul wins.

1997 Poland - Second Round, 5
Tags: probability , number theory , combinatorics , dice , remainder , sum
We have thrown $k$ white dice and $m$ black dice. Find the probability that the remainder modulo $7$ of the sum of the numbers on the white dice is equal to the remainder modulo $7$ of the sum of the numbers on the black dice.

2015 Caucasus Mathematical Olympiad, 4
Tags: digit , remainder , number theory
Is there a nine-digit number without zero digits, the remainder of dividing which on each of its digits is different?

2013 Argentina National Olympiad, 3
Tags: digit , remainder , number theory
Find how many are the numbers of $2013$ digits $d_1d_2…d_{2013}$ with odd digits $d_1,d_2,…,d_{2013}$ such that the sum of $1809$ terms $$d_1 \cdot d_2+d_2\cdot d_3+…+d_{1809}\cdot d_{1810}$$ has remainder $1$ when divided by $4$ and the sum of $203$ terms $$d_{1810}\cdot d_{1811}+d_{1811}\cdot d_{1812}+…+d_{2012}\cdot d_{2013}$$ has remainder $1$ when dividing by $4$.

2007 Thailand Mathematical Olympiad, 15
Tags: remainder , number theory
Compute the remainder when $222!^{111} + 111^{222!} + 111!^{222} + 222^{111!}$ is divided by $2007$.

2016 Indonesia MO, 7
Tags: remainder , sum , number theory
Suppose that $p> 2$ is a prime number. For each integer $k = 1, 2,..., p-1$, denote $r_k$ as the remainder of the division $k^p$ by $p^2$. Prove that $r_1+r_2+r_3+...+r_{p-1}=\frac{p^2(p-1)}{2}$

2000 Estonia National Olympiad, 1
Tags: power of prime , prime , remainder , number theory
Find all prime numbers whose sixth power does not give remainder $1$ when dividing by $504$

2019 Istmo Centroamericano MO, 1
Tags: remainder , digit , number theory
Determine all the numbers formed by three different and non-zero digits, such that the six numbers obtained by permuting these digits leaves the same remainder after the division by $4$.

2006 Thailand Mathematical Olympiad, 13
Tags: number theory , remainder
Compute the remainder when $\underbrace{\hbox{11...1}}_{\hbox{1862}}$ is divided by $2006$

2022 Chile Junior Math Olympiad, 3
Tags: number theory , remainder
By dividing $2023$ by a natural number $m$, the remainder is $23$. How many numbers $m$ are there with this property?

2006 Thailand Mathematical Olympiad, 11
Tags: prime , remainder , product , number theory
Let $p_n$ be the $n$-th prime number. Find the remainder when $\Pi_{n=1}^{2549} 2006^{p^2_{n-1}}$ is divided by $13$

2001 Paraguay Mathematical Olympiad, 2
Tags: remainder , number theory
Find the four smallest four-digit numbers that meet the following condition: by dividing by $2$, $3$, $4$, $5$ or $6$ the remainder is $ 1$.