Found problems: 114
Compute the remainder when $29^{30 }+ 31^{28} + 28! \cdot 30!$ is divided by $29 \cdot 31$.
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 $8$?
Find the remainder after division of the polynomial $x+x^3 +x^9 +x^{27} +x^{81} +x^{243}$ by $x-1$.
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]
For each integer $n > 1$, find a set of $n$ integers $\{a_1, a_2,..., a_n\}$ such that the set of numbers $\{a_1+a_j | 1 \le i \le j \le n\}$ leave distinct remainders when divided by $n(n + 1)/2$. If such a set of integers does not exist, give a proof.
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 $q^2 + r = 1977.$
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 $6$?
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 $k$.
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.
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 $16$.
Consider the set $A = \{1, 2, 3, ..., 2008\}$. We say that a set is of [i]type[/i] $r, r \in \{0, 1, 2\}$, if that set is a nonempty subset of $A$ and the sum of its elements gives the remainder $r$ when divided by $3$. Denote by $X_r, r \in \{0, 1, 2\}$ the class of sets of type $r$. Determine which of the classes $X_r, r \in \{0, 1, 2\}$, is the largest.
Let $a > 2$ be a natural number. Show that there are infinitely many natural numbers n such that $a^n \equiv -1$ (mod $n^2$).
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.
Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.
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 $4$.
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:
$a)$ $n=7$
$b)$ $n=8$
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 $7$.
Find all integers $n$, $n \ge 2$, such that the numbers $1!, 2 !,..., (n - 1)!$ give distinct remainders when divided by $n$.
Find the remainder after division of $10^{10} + 10^{10^2} + 10^{10^3} + ... + 10^{10^{10}}$ by $7$.
For each positive integer $n$, define $s(n) =\sum_{k=0}^n r_k$, where $r_k$ is the remainder when $n \choose k$ is divided by $3$. Find all positive integers $n$ such that $s(n) \ge n$.
Malik Talbi
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 $4$.
Find all pairs $(a, p)$ of positive integers, where $p$ is a prime, such that for any pair of positive integers $m$ and $n$ the remainder obtained when $a^{2^n}$ is divided by $p^n$ is non-zero and equals the remainder obtained when $a^{2^m}$ is divided by $p^m$.
The prime numbers $p$ and $q$ and the integer $a$ are chosen such that $p> 2$ and $a \not\equiv 1$ (mod $q$), but $a^p \equiv 1$ (mod $q$). Prove that $(1 + a^1)(1 + a^2)...(1 + a^{p - 1})\equiv 1$ (mod $q$) .
Source: 1976 Euclid Part B Problem 4
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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 $x-1$.
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 $m$ .