Let $n$ be a non-negative integer. Show that $2^{n+1}$ divides the value of $\left\lfloor{(1+\sqrt{3})^{2n+1}}\right\rfloor$ where function $\lfloor{x}\rfloor$ returns the largest integer not exceeding the give real number $x$.
Let $n$ be a positive integer, show that $(3^{3n}-26n-1)$ is divisible by $676$.
Let $n$ be a positive integer. Show that $\left(3^{4n+2} + 5^{2n+1}\right)$ is divisible by $14$.
A four digit number is divisible by all the even numbers strictly between $10$ and $20$. This four digit number plus the sum of its own digits equals a perfect square. Find this four digit number.
For each positive integer $n > 1$, let $P(n)$ denote the greatest prime factor of $n$. For how many positive integers $n$ is it true that both $P(n) = \sqrt{n}$ and $P(n+48) = \sqrt{n+48}$?
How many perfect squares are divisors of the product $1! \cdot 2! \cdot 3! \cdot \cdots \cdot 9!$?
Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. The number of possible values of $k$ is
Several sets of prime numbers, such as $\{7,83,421,659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
Suppose that $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?
Two players, $A$ and $B$, take turns naming positive integers, with $A$ playing first. No player may name an integer that can be expressed as a linear combination, with positive integer coefficients, of previously named integers. The player who names 1 loses. Show that no matter how A and B play, the game will always end.
Let $A_n$ be the average of all the integers between 1 and 101 which are the multiples of $n$ . Which is the largest among $A_2, A_3, A_4, A_5$ and $A_6$?
A positive integer $n$ is said to be good if there exists a perfect square whose sum of digits in base $10$ is equal to $n$. For instance, $13$ is good because $7^2 = 49$ and $4 + 9 = 13$. How many good numbers are among $1, 2, 3, \cdots , 2007$?
A prime number is called an absolute prime if every permutation of its digits in base 10 is also a prime number. For example: 2, 3, 5, 7, 11, 13 (31), 17 (71), 37 (73) 79 (97), 113 (131, 311), 199 (919, 991) and 337 (373, 733) are absolute primes. Prove that no absolute prime contains all of the digits 1, 3, 7 and 9 in base 10.
Mary found a $3$-digit number that, when multiplied by itself, produced a number which ended in her original $3$-digit number. What is the sum of all the numbers which have this property?
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.
A base-10 three digit number $n$ is selected at random. Which of the following is closest to the probability that the base-9 representation and the base-11 representation of $n$ are both three-digit numerals?
Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $100$. For how many values of $n$ is $q+r$ divisible by $11$?
Find all prime numbers $p$ that can be written $p = x^4 + 4y^4$, where $x, y$ are positive integers.
Is $4^{545} + 545^{4}$ a prime?
Prove that if $n>1$, then $(n^4 + 4^n)$ is a composite number.
Compute $$\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}$$
Find the largest prime divisor of $25^2+72^2$
Calculate the value of $$\dfrac{2014^4+4 \times 2013^4}{2013^2+4027^2}-\dfrac{2012^4+4 \times 2013^4}{2013^2+4025^2}$$
For $k > 0$, let $I_k = 10\ldots 064$, where there are $k$ zeros between the $1$ and the $6$. Let $N(k)$ be the number of factors of $2$ in the prime factorization of $I_k$. What is the maximum value of $N(k)$?
A rectangular box has integer side lengths in the ratio $1: 3: 4$. Which of the following could be the volume of the box?