Found problems: 114
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
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 $q^2 + r = 1977.$
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. $17$
Let $f(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$. Find the remainder when $f(x^7)$ is divided by $f(x)$.
Let $n$ be an integer, $n \ge 2$. Find the remainder of the division of the number $n(n + 1)(n + 2)$ by $n - 1$.
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 $9$.
Let $m$ be a positive integer less than $2015$. Suppose that the remainder when $2015$ is divided by $m$ is $n$. Compute the largest possible value of $n$.
[i] Proposed by Michael Ren [/i]
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 $13$
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 $f(1), f(2), ..., f(2019^3)$.
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 $.
What is the remainder when $\sum_{k=1}^{2005}k^{2005\cdot 2^{2005}}$ is divided by $2^{2005}$?
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]
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
Let $p$ be an odd prime number. Prove that
$$1^{p-1} + 2^{p-1} +...+ (p-1)^{p-1} \equiv p + (p-1)! \mod p^2$$