If $p$, $q$, $\frac{2p-1}{q}$, and $\frac{2q-1}{p}$ are all integers, and $p>1$, $q>1$, find the value of $(p+q)$
Let positive integer $d$ is a divisor of $2n^2$, where $n$ is also a positive integer. Prove $(n^2 + d)$ cannot be a perfect square.
Find all prime number $p$ such that both $(4p^2+1)$ and $(6p^2+1)$ are prime numbers.
Prove there exist infinite number of positive integer $a$ such that for any positive integer $n$, $n^4 + a$ is not a prime number.
What is the smallest positive integer than can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?
Find all positive integer $n$ such that $(3^{2n+1} -2^{2n+1}- 6^n)$ is a composite number.
Find $n$ different positive integers such that any two of them are relatively prime, but the sum of any $k$ ($k < n$) of them is a composite number.
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 $B$ be the set of all binary integers that can be written using exactly $5$ zeros and $8$ ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer $1$ is obtained.
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 $1000$.
Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 12n = 2012$.
Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Find the $1000^{th}$ number in $S$ (in base $10$).
For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \ldots\}$. Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$-safe, $11$-safe, and $13$-safe.
Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$, $x_1$ , $\dots$ , $x_{2011}$ such that \[m^{x_0} = \sum_{k = 1}^{2011} m^{x_k}.\]
Let $R$ be the set of all possible remainders when a number of the form $2^n$, where $n$ is a non-negative integer, is divided by $1000$. Let $S$ be the sum of the elements in $R$. Find the remainder when $S$ is divided by $1000$.
There are $N$ permutations $(a_{1}, a_{2}, ... , a_{30})$ of $1, 2, \ldots, 30$ such that for $m \in \left\{{2, 3, 5}\right\}$, $m$ divides $(a_{n+m} - a_{n})$ for all integers $n$ with $1 \leq n < n+m \leq 30$. Find $N$.
What is the last digit of $9^{2019}$?
What are the last two digits of $8^{88}$?
Find the remainder when $3^{2019} + 4^{2019}$ is divided by 5?
Joe uses 9 different digits out of 0 to 9 to create a 2-digit number, a 3-digit number, and a 4-digit number. He \ffinds the sum of these three numbers is 2014. Do you know which digit is not used?
Solve in integer: $36((xy+1)z+x)=475(yz+1)$
Find the remainder when $9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999}}$ is divided by $1000$.
Let $N$ be the greatest integer multiple of $36$ all of whose digits are even and no two of whose digits are the same. Find the remainder when $N$ is divided by $1000$.
Let $K$ be the product of all factors $(b-a)$ (not necessarily distinct) where $a$ and $b$ are integers satisfying $1\le a < b \le 20$. Find the greatest positive integer $n$ such that $2^n$ divides $K$.