Primes and Factors: all problems

A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?

A single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N$?

What is the least positive integer $m$ such that $m \cdot 2! \cdot 3! \cdot 4! \cdot 5! \cdots 16!$ is a perfect square?

For how many (not necessarily positive) integer values of $n$ is the value of $4000 \cdot \left(\frac{2}{5}\right)^n$ an integer?

The numbers from $1$ to $8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?

How many positive even multiples of $3$ less than $2020$ are perfect squares?

How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)?

What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$?

How many ways are there to write $2016$ as the sum of twos and threes, ignoring order? (For example, $1008 \cdot 2 + 0 \cdot 3$ and $402 \cdot 2 + 404 \cdot 3$ are two such ways.)

Find the number of positive integers with three not necessarily distinct digits, $abc$, with $a \neq 0$ and $c \neq 0$ such that both $abc$ and $cba$ are multiples of $4$.

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?

Let $S$ be the set of all positive integer divisors of $100{,}000$. How many numbers are the product of two distinct elements of $S$?

The $25$ integers from $-10$ to $14$, inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?

All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?

How many odd positive $3$-digit integers are divisible by $3$ but do not contain the digit $3$?

How many positive two-digit integers are factors of $2^{24} - 1$?

There is a prime number $p$ such that $16p + 1$ is the cube of a positive integer. Find $p$.

Joey and Chloe and their daughter Zoe all have the same birthday. Joey is $1$ year older than Chloe, and Zoe is exactly $1$ year old today. Today is the first of the $9$ birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?

What is the tens digit of $2015^{2016} - 2017$?

Let $N = 123456789101112\dots4344$ be the $79$-digit number that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$?

What is the hundreds digit of $(20! - 15!)$?

Find the number of ordered pairs of positive integers $(m, n)$ such that $m^2 n = 20^{20}$.

Let $N = 34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$?

In multiplying two positive integers $a$ and $b$, Ron reversed the digits of the two-digit number $a$. His erroneous product was $161$. What is the correct value of the product of $a$ and $b$?

Positive integers $a$, $b$, and $2009$, with $a < b < 2009$, form a geometric sequence with an integer ratio. What is $a$?

How many positive cubes divide $3! \cdot 5! \cdot 7!$?

Let $N = 34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$?

Suppose that $m$ and $n$ are positive integers such that $75m = n^3$. What is the minimum possible value of $m + n$?

Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at $4$¢ per glob and $J$ blobs of jam at $5$¢ per blob. The cost of the peanut butter and jam to make all the sandwiches is $2.53. Assume that $B$, $J$, and $N$ are positive integers with $N > 1$. What is the cost of the jam Elmo uses to make the sandwiches?

The largest prime factor of $16384$ is $2$ because $16384 = 2^{14}$. What is the sum of the digits of the greatest prime number that is a divisor of $16383$?

How many ways are there to paint each of the integers $2, 3, \ldots, 9$ either red, green, or blue so that each number has a different color from each of its proper divisors?

The positive integers $A$, $B$, $A - B$, and $A + B$ are all prime numbers. The sum of these four primes is

Given that $3^8 \cdot 5^2 = a^b$, where both $a$ and $b$ are positive integers, find the smallest possible value for $a + b$.

How many positive integer divisors of $2004^{2004}$ are divisible by exactly $2004$ positive integers?

Find the number of positive integers that are divisors of at least one of $10^{10}$, $15^7$, $18^{11}$.

Let $S$ be the sum of all numbers of the form $a/b$, where $a$ and $b$ are relatively prime positive divisors of $1000$. What is the greatest integer that does not exceed $S/10$?

For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12$?

For each positive integer $n > 1$, let $P(n)$ denote the greatest prime factor of $n$. For how many positive integers $n$ is it true that both $P(n) = \sqrt{n}$ and $P(n + 48) = \sqrt{n + 48}$?