For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?
How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600$ and $\text{lcm}(y,z)=900$?
Find a square number which has two thousand and eighteen $6$s and some numbers of $0$s?
In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?
There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\gcd(a, b, c, d) = 77$ and $\operatorname{lcm}(a, b, c, d) = n$. What is the smallest possible value for $n$?
How many ordered integeral triples $(x, y, z)$ have the property that each number is the product of the other two?
Let $n$ be a positive integer, and $d$ is a positive divisor of $2n^2$. Show that $(n^2+d)$ cannot be a square number.
Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes.
An $a \times b \times c$ rectangular box is built from $a \cdot b \cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \times b \times c$ parallel to the $(b \times c)$ faces of the box contains exactly $9$ red cubes, exactly $12$ green cubes, and some yellow cubes. Each of the $b$ layers of size $a \times 1 \times c$ parallel to the $(a \times c)$ faces of the box contains exactly $20$ green cubes, exactly $25$ yellow cubes, and some red cubes. Find the smallest possible volume of the box.
For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$-nice nor $8$-nice.
The numbers from 1 to 7 are separated into two non-empty sets A and B. The numbers in A are multiplied together to get a. The numbers in B are multiplied together to get b. The larger of the two numbers a and b is written down. What is the smallest number that can be written down using this procedure?
The sum of three distinct 2-digit primes is 53. Two of the primes have a units digit of 3, and the other prime has a units digit of 7. What is the greatest of the three primes?
If 738 consecutive integers are added together, where the 178th number in the sequence is 4,256,815, what is the remainder when this sum is divided by 6?
Given that $a, b,$ and $c$ are positive integers such that $a^b\cdot b^c$ is a multiple of 2016. Compute the least possible value of $a+b+c$.
Find the largest of three prime divisors of $13^4+16^5-172^2$.
The number $21982145917308330487013369$ is the thirteenth power of a positive integer. Which positive integer?
The number $N$ is a two-digit number.
• When $N$ is divided by $9$, the remainder is $1$.
• When $N$ is divided by $10$, the remainder is $3$.
What is the remainder when $N$ is divided by $11$?
What is the sum of the distinct prime integer divisors of $2016$?
The sum of two prime numbers is 85. What is the product of these two prime numbers?
If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible?
The 7-digit numbers $\underline{7} \underline{4} \underline{A} \underline{5} \underline{2} \underline{B} \underline{1}$ and $\underline{3} \underline{2} \underline{6} \underline{A} \underline{B} \underline{4} \underline{C}$ are each multiples of 3. Which of the following could be the value of $C$?
If $17!=355687ab8096000$ where $a$ and $b$ are two missing single digits. Find $a$ and $b$.
We define the Fibonaccie numbers by $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n}$. Find the greatest common divisor $(F_{100}, F_{99})$, and $(F_{100}, F_{96})$.
A rational number written in base eight is $\underline{a} \underline{b} . \underline{c} \underline{d}$, where all digits are nonzero. The same number in base twelve is $\underline{b} \underline{b} . \underline{b} \underline{a}$. Find the base-ten number $\underline{a} \underline{b} \underline{c}$.
Find the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$.