How many positive integers, not exceeding $2019$, are relatively prime to $2019$?
Let $p$ be a prime number, computer $\varphi(p)$.
Let $p$ be a prime number and $n$ be a positive integer. Show that $\varphi(p^n)=p^n - p^{n-1}$ where $\varphi(n)$ is the Euler's totient function.
Show that if $a$ and $b$ are relatively prime, then $\varphi(a)\varphi(b)=\varphi(ab)$ where $\varphi(n)$ is Euler's totient function.
Twin primes are prime numbers that differ by 2. Given that $a$ and $b$ are the greatest twin primes with $a < 100$, evaluate the value of $a + b$.
Find the number of ending zeros of $2014!$ in base 9. Give your answer in base 9.
Find the number of fractions in the following list that is in its lowest form (i.e. the denominator and the numerator are co-prime). $$\frac{1}{2014}, \frac{2}{2013}, \frac{3}{2012}, \cdots, \frac{1007}{1008}$$
On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process continues through June 12 with a different number of students per row each day. However, on June 13, they cannot find a new way of organizing the students. What is the smallest possible number of students in the group?
A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N>2M$ and $M>4$. Each team plays a 76 game schedule. How many games does a team play within its own division?
Let $m$ and $n$ be two positive integers between $2$ and $99$, inclusive. Mr. $S$ knows their sum, and Mr. $P$ knows their product. Following are their conversations:
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Mr. $S$: I am certain that you don't know these two numbers individually. But I don't know them either.
- Mr. $P$: Yes, I didn't know. But I know them now.
- Mr. $S$: If this is the case, I know them now too.
What are the two numbers?
What is the $22^{nd}$ positive integer $n$ such that $22^n$ ends in a $2$?
Find the sum of all positive integers $n$ such that the least common multiple of $2n$ and $n^2$ equals $(14n - 24)$?
What is the largest positive integer $n$ less than $10,000$ such that in base 4, $n$ and $3n$ have the same number of digits; in base 8, $n$ and $7n$ have the same number of digits; and in base 16, $n$ and $15n$ have the same number of digits? Express your answer in base 10.
For every integer $n$, let $m$ denote the integer made up of the last four digit of $n^{2015}$. Consider all positive integer $n < 10000$, let $A$ be the number of cases when $n > m$, and $B$ be the number of cases when $n < m$. Compute $A-B$.
Let $\frac{p}{q}=1+ \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{100000}$ where $p$ and $p$ are both positive integers and do not have common divisor greater than 1. How many ending zeros does $q$ have?
Let $m$ be a positive odd integer, $m\ge 2$. Find the smallest positive integer $n$ such that $2^{2015}$ divides $m^n-1$.
Find the largest 7-digit integer such that all its 3-digit subpart is either a multiple of 11 or multiple of 13.
Find a $4$-digit square number $x$ such that if every digit of $x$ is increased by 1, the new number is still a perfect square.
Show that the next integer bigger than $(\sqrt{2}+1)^{2n}$ is divisible by $2^{n+1}$.
Let $m$ be an odd positive integer, and not a multiple of 3. Show that the integer part of $4^m - (2+\sqrt{2})^m$ is a multiple of 112.
If the first $25$ positive integers are multiplied together, in how many zeros does the product terminate?
Show that the sum of all the numbers of the form $\frac{1}{mn}$ is not an integer, where $m$ and $n$ are integers, and $1\le m \le n \le 2017$.
Show that $1^{2017}+2^{2017}+\cdots + n^{2017}$ is not divisible by $(n+2)$ for any positive integer $n$.
How many integers $m$ are there for which $5\times 2^m +1$ is a square number?
This four digit number $n$ has 14 positive factors and one of its prime factor has last digit equal to 1. What is $n$?