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

1969 Dutch Mathematical Olympiad, 1
Tags: number theory , remainder
Determine the smallest $n$ such that $n \equiv (a - 1)$ mod $a$ for all $a \in \{2,3,..., 10\}$.

2020 OMMock - Mexico National Olympiad Mock Exam, 2
Tags: permutation , number theory , remainder
We say that a permutation $(a_1, \dots, a_n)$ of $(1, 2, \dots, n)$ is good if the sums $a_1 + a_2 + \dots + a_i$ are all distinct modulo $n$. Prove that there exists a positive integer $n$ such that there are at least $2020$ good permutations of $(1, 2, \dots, n)$. [i]Proposed by Ariel García[/i]

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

2009 Postal Coaching, 4
Tags: digit , remainder , number theory
A four - digit natural number which is divisible by $7$ is given. The number obtained by writing the digits in reverse order is also divisible by $7$. Furthermore, both the numbers leave the same remainder when divided by $37$. Find the 4-digit number.

1949-56 Chisinau City MO, 8
Tags: number theory , sum of squares , remainder
Prove that the remainder of dividing the sum of two squares of integers by $4$ is different from $3$.

2009 Cuba MO, 1
Tags: number theory , remainder , prime
Show that when a prime number is divided by $30$, the remainder is $1$ or a prime number. Shows that if it is divided by $60$ or $90$ the same thing does not happen.

2015 Caucasus Mathematical Olympiad, 1
Tags: number theory , divisible , remainder , digit
Is there an eight-digit number without zero digits, which when divided by the first digit gives the remainder $1$, when divided by the second digit will give the remainder $2$, ..., when divided by the eighth digit will give the remainder $8$?

2001 Paraguay Mathematical Olympiad, 2
Tags: number theory , remainder
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$.

2012 Bosnia And Herzegovina - Regional Olympiad, 3
Tags: remainder , number theory
Find remainder when dividing upon $2012$ number $$A=1\cdot2+2\cdot3+3\cdot4+...+2009\cdot2010+2010\cdot2011$$

2025 Kosovo National Mathematical Olympiad`, P4
Tags: remainder , number theory
When a number is divided by $2$ it has quotient $x$ and remainder $1$. Whereas, when the same number is divided by $3$ it has quotient $y$ and remainder $2$. What is the remainder when $x+y$ is divided by $5$?

1994 North Macedonia National Olympiad, 1
Tags: sum , number theory , 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 $6$.

1947 Moscow Mathematical Olympiad, 123
Tags: remainder , polynomial , algebra
Find the remainder after division of the polynomial $x+x^3 +x^9 +x^{27} +x^{81} +x^{243}$ by $x-1$.

1977 IMO Shortlist, 3
Tags: number theory , additive number theory , remainder , divisibility
Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$

1946 Moscow Mathematical Olympiad, 108
Tags: number theory , 4-digit , remainder
Find a four-digit number such that the remainders after its division by $131$ and $132$ are $112$ and $98$, respectively.

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?

2004 IMO Shortlist, 6
Tags: modular arithmetic , number theory , divisibility , remainder
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]

2014 Junior Regional Olympiad - FBH, 4
Tags: number theory , remainder
Positive integer $n$ when divided with number $3$ gives remainder $a$, when divided with $5$ has remainder $b$ and when divided with $7$ gives remainder $c$. Find remainder when dividing number $n$ with $105$ if $4a+3b+2c=30$

1990 ITAMO, 3
Tags: polynomial , remainder , algebra
Let $a,b,c$ be distinct real numbers and $P(x)$ a polynomial with real coefficients. Suppose that the remainders of $P(x)$ upon division by $(x-a), (x-b)$ and $(x-c)$ are $a,b$ and $c$, respectively. Find the polynomial that is obtained as the remainder of $P(x)$ upon division by $(x-a)(x-b)(x-c)$.

2017 Dutch Mathematical Olympiad, 4
Tags: remainder , sum , number theory
If we divide the number $13$ by the three numbers $5, 7$, and $9$, then these divisions leave remainders: when dividing by $5$ the remainder is $3$, when dividing by $7$ the remainder is $6$, and when dividing by $9$ the remainder is 4. If we add these remainders, we obtain $3 + 6 + 4 = 13$, the original number. (a) Let $n$ be a positive integer and let $a$ and $b$ be two positive integers smaller than $n$. Prove: if you divide $n$ by $a$ and $b$, then the sum of the two remainders never equals $n$. (b) Determine all integers $n > 229$ having the property that if you divide $n$ by $99, 132$, and $229$, the sum of the three remainders is $n$.

2019 Istmo Centroamericano MO, 1
Tags: number theory , remainder , digit
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 $4$.

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 $20$ ?

2020 New Zealand MO, 2
Tags: number theory , remainder
Find the smallest positive integer $N$ satisfying the following three properties. $\bullet$ N leaves a remainder of $5$ when divided by $7$. $\bullet$ N leaves a remainder of $6$ when divided by $ 8$. $\bullet$ N leaves a remainder of $7$ when divided by $9$.

1990 Bundeswettbewerb Mathematik, 1
Tags: remainder , number theory
Find all triples $(a,b,c)$ of positive integers such that the product of any two of them when divided by the third leaves the remainder $1$.

2015 Singapore Senior Math Olympiad, 3
Tags: number theory , remainder
Let $n \ge 3$ be an integer. Prove that there exist positive integers $\ge 2$, $a_1,a_2,..,a_n$, such that $a_1 a_2 ... \widehat{a_i}... a_n \equiv 1$ (mod $a_i$), for $i = 1,..., n$. Here $\widehat{a_i}$ means the term $a_i$ is omitted.

2022 Switzerland - Final Round, 2
Tags: number theory , remainder , divide
Let $n$ be a positive integer. Prove that the numbers $$1^1, 3^3, 5^5, ..., (2n-1)^{2n-1}$$ all give different remainders when divided by $2^n$.