The smallest number greater than $2$ that leaves a remainder of $2$ when divided by $3$, $4$, $5$, or $6$ lies between what numbers?
What are the sign and units digit of the product of all the odd negative integers strictly greater than $-2015$?
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?
What is the hundreds digit of $2011^{2011}?$
The number obtained from the last two non-zero digits of $90!$ is equal to $n$. What is $n$?
What is the remainder when $3^0 + 3^1 + 3^2 + \cdots + 3^{2009}$ is divided by $8$?
Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?
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 $5$?
What is the tens digit in the sum $7!+8!+9!+...+2018!$
What is the units digit of the product $7^{23} \times 8^{105} \times 3^{18}$?
When $(37 \times 45) - 15$ is simplified, what is the units digit?
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 there is no integer solutions to $x^2 = y^5 - 4$.
Find the smallest positive integer $n$ such that the last $3$ digits of $n^3$ is $888$.
Solve the congruence $5x \equiv 21 \pmod{37}$.
Show that $n^{13} \equiv n\pmod{2730}$ for all integers $n$.
Let $f(n)$ denote the sum of the digits of $n$. Find $f(f(f(4444^{4444})))$.
Prove that if $p$ and $(p^2 + 8)$ are prime, then $(p^3 + 8p + 2)$ is prime.
Prove that for every prime $p$, there exists an integer $x$, such that $x^8 \equiv 16 \pmod{p}$
How many among the first $1000$ Fibonacci numbers are multiples of $11$?
Show that for any right triangle whose sides' lengths are all integers,
- one side's length must be a multiple of 3, and
- one side's length must be a multiple of 4, and
- one side's length must be a multiple of 5
Please note these sides may not be distinct. For example, in a 5-12-13 triangle, 12 is a multiple of both 3 and 4.
Let $p$ be a prime. Prove that the equation $x^2-py^2 = -1$ has integral solution if and only if $p=2$ or $p\equiv 1\pmod{4}$.