Divisibility & Legendre's Formula: all problems

A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?

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

For all integers $n \ge 9$, the value of

is always which of the following?

Prove that the fraction $\frac{21n + 4}{14n + 3}$ is irreducible for every natural number $n$.

What is the largest positive integer $n$ for which $n^3 + 100$ is divisible by $n + 10$?

Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357$, $89$, $5$ are all uphill integers but $32$, $1240$, $466$ are not. How many uphill integers are divisible by $15$?

What is the largest integer that is a divisor of

for all positive even integers $n$?

The number $21! = 51{,}090{,}942{,}171{,}709{,}440{,}000$ has over $60{,}000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?

Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357$, $89$, and $5$ are all uphill integers, but $32$, $1240$, and $466$ are not. How many uphill integers are divisible by $15$?

Let $N = 123456789101112 \dots 4344$ 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 $45$?

Let $n$ denote the smallest positive integer that is divisible by both $4$ and $9$, and whose base-10 representation consists of only $4$'s and $9$'s, with at least one of each. What are the last four digits of $n$?

A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms $247$, $475$, and $756$ and end with the term $824$. Let $S$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $S$?

For how many positive integers $n \le 1000$ is

not divisible by $3$? (Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)

Let $a_{10} = 10$, and for each positive integer $n > 10$ let $a_n = 100a_{n-1} + n$. Find the least positive $n > 10$ such that $a_n$ is a multiple of $99$.

Define $n!!$ to be $n(n - 2)(n - 4) \cdots 3 \cdot 1$ for $n$ odd and $n(n - 2)(n - 4) \cdots 4 \cdot 2$ for $n$ even. When

is expressed as a fraction in lowest terms, its denominator is $2^a b$ with $b$ odd. Find $\dfrac{ab}{10}$.