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

2025 Kosovo National Mathematical Olympiad`, P3
Tags: number theory , prime , remainder
Let $m$ and $n$ be natural numbers such that $m^3-n^3$ is a prime number. What is the remainder of the number $m^3-n^3$ when divided by $6$?

1963 German National Olympiad, 1
Tags: divide , number theory , remainder
a) Prove that when you divide any prime number by $30$, the remainder is either $1$ or is a prime number! b) Does this also apply when dividing a prime number by $60$? Justify your answer!

2015 Bosnia and Herzegovina Junior BMO TST, 4
Tags: combinatorics , positive integer , remainder , sum
Let $n$ be a positive integer and let $a_1$, $a_2$,..., $a_n$ be positive integers from set $\{1, 2,..., n\}$ such that every number from this set occurs exactly once. Is it possible that numbers $a_1$, $a_1 + a_2 ,..., a_1 + a_2 + ... + a_n$ all have different remainders upon division by $n$, if: $a)$ $n=7$ $b)$ $n=8$

2024 AMC 10, 7
Tags: remainder , modular arithmetic
What is the remainder when $7^{2024}+7^{2025}+7^{2026}$ is divided by $19$? $ \textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }7 \qquad \textbf{(D) }11 \qquad \textbf{(E) }18 \qquad $

1981 Poland - Second Round, 4
Tags: number theory , remainder
The given natural numbers are $ k, n $. We inductively define two sequences of numbers $ (a_j) $ and $ (r_j) $ as follows: Step one: we divide $ k $ by $ n $ and get the quotient $ a_1 $ and the remainder $ r_i $, step j: we divide $ k+r_{j-1} $ by $ n $ and get the quotient $ a_j $ and the remainder $ r_j $. Calculate the sum of $ a_1 + \ldots + a_n $.

2022 IFYM, Sozopol, 5
Tags: remainder , number theory
Find the number of subsets of $\{1, 2,... , 2100\}$ such that each has sum of the elements giving a remainder of $3$ when divided by $7$.

1949-56 Chisinau City MO, 6
Tags: number theory , perfect square , remainder
Prove that the remainder of dividing the square of an integer by $3$ is different from $2$.

2021 Saudi Arabia Training Tests, 32
Tags: number theory , algebra , sequence , remainder
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 $n \ge M$.

2004 Thailand Mathematical Olympiad, 13
Tags: remainder , number theory
Compute the remainder when $29^{30 }+ 31^{28} + 28! \cdot 30!$ is divided by $29 \cdot 31$.

2024 AMC 10, 18
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 $

2013 Saudi Arabia GMO TST, 4
Tags: fibonacci number , remainder , number theory
Let $F_0 = 0, F_1 = 1$ and $F_{n+1} = F_n + F_{n-1}$, for all positive integer $n$, be the Fibonacci sequence. Prove that for any positive integer $m$ there exist infinitely many positive integers $n$ such that $F_n + 2 \equiv F_{n+1} + 1 \equiv F_{n+2}$ mod $m$ .

2018 Hanoi Open Mathematics Competitions, 8
Tags: number theory , remainder
Let $k$ be a positive integer such that $1 +\frac12+\frac13+ ... +\frac{1}{13}=\frac{k}{13!}$. Find the remainder when $k$ is divided by $7$.

1976 Euclid, 4
Tags: remainder , polynomial division , algebra , polynomial
Source: 1976 Euclid Part B Problem 4 ----- The remainder when $f(x)=x^5-2x^4+ax^3-x^2+bx-2$ is divided by $x+1$ is $-7$. When $f(x)$ is divided by $x-2$ the remainder is $32$. Determine the remainder when $f(x)$ is divided by $x-1$.

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.

2010 District Olympiad, 2
Tags: remainder , number theory
Let $n$ be an integer, $n \ge 2$. Find the remainder of the division of the number $n(n + 1)(n + 2)$ by $n - 1$.

1999 Estonia National Olympiad, 1
Tags: number theory , remainder , power of 2 , power of 3
Let $a, b, c$ and $d$ be non-negative integers. Prove that the numbers $2^a7^b$ and $2^c7^d$ give the same remainder when divided by $15$ iff the numbers $3^a5^b$ and $3^c5^d$ give the same remainder when divided by $16$.

2005 Thailand Mathematical Olympiad, 10
Tags: number theory , remainder , sum
What is the remainder when $\sum_{k=1}^{2005}k^{2005\cdot 2^{2005}}$ is divided by $2^{2005}$?

2019 Taiwan APMO Preliminary Test, P2
Tags: remainder , modulo , number theory
Put $1,2,....,2018$ (2018 numbers) in a row randomly and call this number $A$. Find the remainder of $A$ divided by $3$.

2009 Ukraine Team Selection Test, 6
Tags: modulo , number theory , remainder , set
Find all odd prime numbers $p$ for which there exists a natural number $g$ for which the sets \[A=\left\{ \left( {{k}^{2}}+1 \right)\,\bmod p|\,k=1,2,\ldots ,\frac{p-1}{2} \right\}\] and \[B=\left\{ {{g}^{k}}\bmod \,p|\,k=1,2,...,\frac{p-1}{2} \right\}\] are equal.

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 $f(1), f(2), ..., f(2019^3)$.

2006 Thailand Mathematical Olympiad, 10
Tags: number theory , factorial , remainder
Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.

2018 India PRMO, 25
Tags: number theory , sum , remainder , digit
Let $T$ be the smallest positive integers which, when divided by $11,13,15$ leaves remainders in the sets {$7,8,9$}, {$1,2,3$}, {$4,5,6$} respectively. What is the sum of the squares of the digits of $T$ ?

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

1963 Swedish Mathematical Competition., 3
Tags: number theory , remainder
What is the remainder on dividing $1234^{567} + 89^{1011}$ by $12$?

1977 IMO, 2
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 $q^2 + r = 1977.$