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Let $a > b > c$ be three positive integers. If their remainders are $2$, $7$, and $9$ respectively when being divided by $11$. Find the remainder when $(a+b+c)(a-b)(b-c)$ is divided by $11$.

Find all positive integer $n$ such that $2^n+1$ is divisible by 3.

What is the remainder when $\left(8888^{2222} + 7777^{3333}\right)$ is divided by $37$?

Find all prime number $p$ such that both $(4p^2+1)$ and $(6p^2+1)$ are prime numbers.

Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$, of play blocks which satisfies the conditions:

• If $16$, $15$, or $14$ students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and
• There are three integers $0 < x < y < z < 14$ such that when $x$, $y$, or $z$ students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.

Find the sum of the distinct prime divisors of the least possible value of $N$ satisfying the above conditions.

For positive integers $n$ and $k$, let $f(n, k)$ be the remainder when $n$ is divided by $k$, and for $n > 1$ let $F(n) =\displaystyle\max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)$. Find the remainder when $\sum\limits_{n=20}^{100} F(n)$ is divided by $1000$.

Let $\mathcal{S}$ be the set of all perfect squares whose rightmost three digits in base $10$ are $256$. Let $\mathcal{T}$ be the set of all numbers of the form $\frac{x-256}{1000}$, where $x$ is in $\mathcal{S}$. In other words, $\mathcal{T}$ is the set of numbers when the last three digits of each number in $\mathcal{S}$ are truncated. Find the remainder when the tenth smallest element of $\mathcal{T}$ is divided by $1000$.

Let integer $a$, $b$, and $c$ satisfy $a+b+c=0$, prove $|a^{1999}+b^{1999}+c^{1999}|$ is a composite number.

The number $2017$ is prime. Let $S = \sum \limits_{k=0}^{62} \dbinom{2014}{k}$. What is the remainder when $S$ is divided by $2017$?

A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people?

Let four positive integers $a$, $b$, $c$, and $d$ satisfy $a+b+c+d=2019$. Prove $\left(a^3+b^3+c^3+d^3\right)$ cannot be an even number.

The smallest number greater than $2$ that leaves a remainder of $2$ when divided by  $3$, $4$, $5$, or $6$ lies between what numbers?

Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?

When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over?

Seven students count from $1$ to $1000$ as follows:

• Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says $1$, $3$, $4$, $6$, $7$, $9$, . . ., $997$, $999$, $1000$.
• Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.
• Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.
• Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.
• Finally, George says the only number that no one else says.

What number does George say?

Six distinct positive integers are randomly chosen between $1$ and $2020$, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of $5$?

Farmer Hank has fewer than $100$ pigs on his farm. If he groups the pigs five to a pen, there are always three pigs left over. If he groups the pigs seven to a pen, there is always one pig left over. However, if he groups the pigs three to a pen, there are no pigs left over. What is the greatest number of pigs that Farmer Hank could have on his farm?

$N$ delegates attend a round-table meeting, where $N$ is an even number. After a break, these delegates randomly pick a seat to sit down again to continue the meeting. Prove that there must exist two delegates so that the number of people sitting between them is the same before and after the break.

Prove that if $p$ and $(p^2 + 8)$ are prime, then $(p^3 + 8p + 2)$ is prime.

What is the smallest positive integer greater than $5$ which leaves a remainder of $5$ when divided by each of $6$, $7$, $8$, and $9$?

Show that all the terms of the sequence $a_n=\frac{(2+\sqrt{3})^n-(2-\sqrt{3})^n}{2\sqrt{3}}$ are integers, and also find all the $n$ such that $3 \mid a_n$.

What is the remainder when $2021^{2020}$ is divided by $10^4$?

Find the smallest positive integer $n$ so that $107n$ has the same last two digits as $n$.

Find the largest positive integer $n$ such that $(3^{1024} - 1)$ is divisible by $2^n$.

Find $8$ prime numbers, not necessarily distinct such that the sum of the squares of these numbers is $992$ less than $4$ times of the product of these numbers.