Found problems: 114
Finding all quads of integers $(a, b, c, p)$ where $p \ge 5$ is prime number such that the remainders of the numbers $am^3 + bm^2 + cm$, $m = 0, 1, . . . , p - 1$, upon division of $p$ are two by two different..
Benedek scripted a program which calculated the following sum: $1^1+2^2+3^3+. . .+2021^{2021}$. What is the remainder when the sum is divided by $35$?
Let $n$ be a positive integer. Show that every divisors of $2n^2 - 1$ gives a different remainder after division by $2n$.
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 $5$?
Prove that the remainder after division of the square of any prime $p > 3$ by $12$ is equal to $1$.
A set of numbers $a_1, a_2 , . . . , a_{100}$ is obtained by rearranging the numbers $1 , 2,..., 100$ . Form the numbers
$b_1=a_1$
$b_2= a_1 + a_2$
$b_3=a_1 + a_2 + a_3$
...
$b_{100}=a_1 + a_2 + ...+a_{100}$
Prove that among the remainders on dividing the numbers by $100 , 11$ of them are different .
( L . D . Kurlyandchik , Leningrad)
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 $r_1+r_2+r_3+...+r_{p-1}=\frac{p^2(p-1)}{2}$
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 $n$.
Let $p_k$ be the $k$-th prime number. Find the remainder when $\sum_{k=2}^{2550}p_k^{p_k^4-1}$ is divided by $2550$.
Let $p \ge 3$ be a prime. For $j = 1,2 ,... ,p - 1$, let $r_j$ be the remainder when the integer $\frac{j^{p-1}-1}{p}$ is divided by $p$.
Prove that $$r_1 + 2r_2 + ... + (p - 1)r_{p-1} \equiv \frac{p+1}{2} (\mod p)$$
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
$
Find the remainder upon division by $x^2-1$ of the determinant
$$\begin{vmatrix}
x^3+3x & 2 & 1 & 0
\\ x^2+5x & 3 & 0 & 2
\\x^4 +x^2+1 & 2 & 1 & 3
\\x^5 +1 & 1 & 2 & 3
\\ \end{vmatrix}$$
Prove that the remainder of dividing the square of an integer by $3$ is different from $2$.
Let $A$ be a set of $65$ integers with pairwise different remainders modulo $2016$. Prove that exists a subset $B=\{a,b,c,d\}$ of set $A$ such that $a+b-c-d$ is divisible with $2016$
How many ways are there to permute the first $n$ positive integers such that in the permutation, for each value of $k \le n$, the first $k$ elements of the permutation have distinct remainder mod $k$?
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!
In a sequence of natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_ {1999} $, $ a_n-a_ {n-1} -a_ {n-2} $ is divisible by $ 100 (3 \leq n \leq 1999) $. It is known that $ a_1 = 19$ and $ a_2 = 99$. Find the remainder of $ a_1 ^ 2 + a_2 ^ 2 + \dots + a_ {1999} ^ 2 $ by $8$.
Let be given a positive integer $n \ge 3$. Consider integers $a_1,a_2,...,a_n >1$ with the product equals to $A$ such that: for each $k \in \{1, 2,..., n\}$ then the remainder when $\frac{A}{a_k}$ divided by $a_k$ are all equal to $r$. Prove that $r \le n- 2$
Let $n$ be a positive integer less than $1000$. The remainders obtained when dividing $n$ by $2, 2^2, 2^3, ... , 2^8$, and $2^9$ , are calculated. If the sum of all these remainders is $137$, what are all the possible values of $n$?
Find remainder when dividing upon $2012$ number $$A=1\cdot2+2\cdot3+3\cdot4+...+2009\cdot2010+2010\cdot2011$$
By dividing $2023$ by a natural number $m$, the remainder is $23$. How many numbers $m$ are there with this property?
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 $c = 4$.
If we divide number $19250$ with one number, we get remainder $11$. If we divide number $20302$ with the same number, we get the reamainder $3$. Which number is that?
We call a positive integer [i]interesting [/i] if the number and the number with its digits written in reverse order both leave remainder $2$ in division by $4$.
a) Determine if $2018$ is an interesting number.
b) For every positive integer $n$, find how many interesting $n$-digit numbers there are.
Put $1,2,....,2018$ (2018 numbers) in a row randomly and call this number $A$. Find the remainder of $A$ divided by $3$.