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.
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 $11$?
Let $P(x)$ be a polynomial with integer coefficients satisfying that both $P(0)$ and $P(1)$ are odd. Show that $P(x)$ has no integer zeros.
What is the tens digit of $2015^{2016}-2017?$
Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes.
If $2016$ consecutive integers are added together, where the $999^{th}$ number in the sequence is $1,244,584$, what is the remainder when this sum is divided by $6$?
Prove that $7\mid 8^n-1$ for $n\ge 1$.
Show that $5\mid 4^{2n}-1$ for $n\ge 1$.
Prove that $15\mid 4^{2n}-1$ for $n\ge 1$.
Prove that for every positive integer $n$, there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
Show that if $n$ is an integer greater than $1$, then $(2^n-1)$ is not divisible by $n$.
Show that every integer $k > 1$ has a multiple which is less than $k^4$ and can be written in base 10 using at most 4 different digits.
Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$?
Show that neither $385^{97}$ nor $366^{17}$ can be expressed as the sum of cubes of some consecutive integers.
An integer $N$ is selected at random in the range $1\leq N \leq 2020$. What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?
Last year Isabella took $7$ math tests and received $7$ different scores, each an integer between $91$ and $100$, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was $95$. What was her score on the sixth test?
When each of $702$, $787$, and $855$ is divided by the positive integer $m$, the remainder is always the positive integer $r$. When each of $412$, $722$, and $815$ is divided by the positive integer $n$, the remainder is always the positive integer $s \neq r$. Fine $m+n+r+s$.
For a positive integer $n$, let $d_n$ be the units digit of $1 + 2 + \dots + n$. Find the remainder when \[\sum_{n=1}^{2017} d_n\]is divided by $1000$.
Let $x$ be an integer and $p$ is a prime divisor of $(x^6 + x^5 + \cdots + 1)$. Show that $p=7$ or $p\equiv 1\pmod{7}$.
Prove that there are infinitely many distinct pairs $(a,b)$ of relatively prime positive integers $a > 1$ and $b > 1$ such that $(a^b + b^a)$ is divisible by $(a + b)$.
There are three piles which contain $8$, $9$, and $19$ stones, respectively. You are allowed to choose two piles and transfer one stone from each of them to the third pile. Is it possible to make all piles all contain exactly $12$ stones after several such operations?
Let $p$ be an odd prime number. For positive integer $k$ satisfying $1\le k\le p-1$, the number of divisors of $k p+1$ between $k$ and $p$ exclusive is $a_k$. Find the value of $a_1+a_2+\ldots + a_{p-1}$.
Let $p$ be an odd prime divisor of number $(a^2+1)$ where $a$ is an integer. Show that $p\equiv 1\pmod{4}$.