The sum of the three different positive unit fractions is $\frac{6}{7}$. What is the least number that could be the sum of the denominators of these fractions?
The sum of $n$ consecutive positive integers is 100. What is the greatest possible value of $n$?
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 $a + b$.
The sides of a right triangle all have lengths that are whole numbers. The sum of the length of one leg and the hypotenuse is 49. Find the sum of all the possible lengths of the other leg.
(A) 7 (B) 49 (C) 63 (D) 71 (E) 96
What is the last digit of $17^{17^{17^{17}}}$?
Find the number of ending zeros of $2014!$ in base 9. Give your answer in base 9.
Find the sum of all positive integer $x$ such that $3\times 2^x = n^2-1$ for some positive integer $n$.
Find the number of pairs of integer solution $(x, y)$ that satisfies the equation $$(x-y + 2)(x-y-2) =-(x-2)(y-2)$$
Given $S = \{2, 5, 8, 11, 14, 17, 20,\cdots\}$. Given that one can choose $n$ different numbers from $S$, $\{A_1, A2,\cdots A_n\}$, s.t. $\displaystyle\sum_{i=1}^{n}\frac{1}{A_i}=1$ Find the minimum possible value of $n$.
Find the number of positive integers $n\le 2014$ such that there exists integer $x$ that satisfies the condition that $\displaystyle\frac{x + n}{x-n}$ is an odd perfect square.
Find all number sets $(a, b, c, d)$ s.t. $1 < a \le b \le c \le d, a,b,c,d \in\mathbb{N}$, and $a^2 + b + c + d,
a + b^2 + c + d, a + b + c^2 + d$ and $a + b + c + d^2$ are all square numbers. Sum the value of $d$ across all solution $set(s)$.
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.
What is the smallest positive integer $n$ such that $20\equiv n^{15} \pmod{29}$?
Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ with $p$ prime, $3\le p<100$, and $1\le a < p$ such that the sum $a+a^2+a^3+ \cdots + a^{(p-2)!}$ is not divisible by $p$?
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?
If for any integer $k\ne 27$ and $\big(a-k^{2015}\big)$ is divisible by $(27-k)$, what is the last two digits of $a$?
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.