How many positive integers less than $1998$ are relatively prime to $1547$?
For a positive integer m, we define $m$ as a $\textit{factorial}$ number if and only if there exists a positive integer $k$ for which $m = k\cdot(k - 1)\cdots 2\cdot 1$. We define a positive integer $n$ as a $\textit{Thai}$ number if and only if $n$ can be written as both the sum of two factorial numbers and the product of two factorial numbers.
What is the sum of the five smallest $\textit{Thai}$ numbers?
If integer d is not equal to 2,5 or 13. Prove: there must exist two different elements $a$ and $b$ in set {2, 5, 3, d} such that $ab -
Find all 3-digit integer $\overline{abc}$ that satisfy $\overline{abc} = (a + b + c)^3$.
Find all positive integer $1\le n \le 100$ such that sum of all its digits divides $n$ itself.
Express the golden ratio using a continued fraction.
Find the rational number $p/q$ closest to $\sqrt{\pi}$ wich $q \le 25$.
Find the number $x = [1, 2, 3, 1, 2, 3, \cdots]$. (continued fraction)
Find all nonnegative integers $x$ and $y$ such that $x^3+y^3 = (x+y)^2$.
Find the number of paris $(a, b)$ of nonnegative integers that satisfy $6a+7b=1000$
Solve the congruence $5x \equiv 21 \pmod{37}$.
Show that $n^{13} \equiv n\pmod{2730}$ for all integers $n$.
Prove that the sum of $n$ consecutive perfect squares cannot be a perfect square for $n=3, 4, 5,$ and $6$.
Let $n$ be a positive integer that is one less than a multiple of 24. Prove that if $a$ and $b$ are positive integers such that $ab=n$, then $a+b$ is a multiple of 24.
Find the smallest positive integer $n$ such that the remainder is always $1$ when $n$ is divided by $2$, $3$, $4$, $5$, or $6$. In addition, $n$ must be a multiple of $7$.
Show that any positive integer can be expressed as a sum of integers which are some power of 3.
In how many different ways can 900 be expressed as the product of two (possibly equal) positive integers? Regard $m\cdot n$ and $n\cdot m$ as the same.
A binary palindrome is a positive integer whose standard base 2 (binary) representation is a palindrome (reads the same backward or forward). (Leading zeros are not permitted in the standard representation.) For example, 2015 is a binary palindrome, because in base 2 it is 11111011111. How many positive integers less than 2015 are binary palindromes?
In baseball, a player's batting average is the number of hits divided by the number of at bats, rounded to three decimal places. Danielle's batting average is $0.399$. What is the fewest number of at bets that Danielle could have?
Let $n$ be a positive integer. When the leftmost digit of (the standard base 10 representation of) $n$ is shifted to the rightmost position (the units position), the result is $n/3$. Find the smallest possible value of the sum of the digits of $n$.
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.
Show that if $k \ge 4$, then $lcm(1; 3;\cdots; 2k- 3; 2k- 1) > (2k + 1)^2$ where $lcm$ stands for least common multiple.
Find all positive integers $n$ such that for all odd integers $a$. If $a^2\le n$, then $a|n$.
Find all $n \in \mathbb{Z}^+$ such that $2^n + n | 8^n + n$.