Assorted divisibility puzzles

Odd reciprocals with terminating decimals ●●○○○ 20 ptsIf $\frac{a}{b}$ is the probability that the reciprocal of a randomly selected positive odd integer less than $2010$ gives a terminating decimal, with $a$ and $b$ relatively prime positive integers, find $a+b$ .
Positive integers giving a hole ●●○○○ 20 ptsFind the number of positive integers $n$ for which the graph of $y = \frac{(x + 3)(x - 7)^5}{(x - 1)(x - 7)^n}$ has a hole at $x = 7$ .
Smallest multiple beyond 2010 with terminating decimals ●●○○○ 20 ptsLet $S$ be the set of all integers $k$ such that $\frac{17k}{66}$ and $\frac{13k}{105}$ are both terminating decimals. What is the smallest integer in $S$ that is greater than 2010?
Minimum distinct values in a ratio chain ●●●○○ 50 ptsLet $a_{1}, a_{2}, \ldots, a_{2006}$ be positive integers (not necessarily distinct) such that the 2005 ratios $\frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \ldots, \frac{a_{2005}}{a_{2006}}$ are all distinct. What is the least possible number of different values among $a_{1}, a_{2}, \ldots, a_{2006}$ ?
Least n with many unit-fraction solutions ●●●○○ 50 ptsLet $x$ , $y$ and $n$ be positive integers, and count the solutions $(x, y)$ of $\frac{1}{x}+\frac{1}{y}=\frac{1}{n}$ with $x \le y$ . For $n=4$ there are exactly three such solutions. Find the least value of $n$ for which the number of solutions exceeds one thousand.
Natural numbers with terminating quotients ●●●○○ 50 ptsFind all natural numbers $n \geq 2$ such that for every positive integer $k < n$ , the quotient $\frac{n}{k}$ has a finite decimal representation (or is a whole number). List the values in increasing order.
Integers reaching one in twenty steps ●●●●○ 100 ptsFor every positive integer $x$ , let $f(x)=\begin{cases}1 & \text{if }x = 1\\ \frac x{10} & \text{if }x\text{ is divisible by }10\\ x+1 & \text{otherwise}\end{cases}$ and define a sequence by $x_1=x$ and $x_{n+1}=f(x_n)$ for all positive integers $n$ . Let $d(x)$ be the smallest $n$ such that $x_n=1$ . (For example, $d(100)=3$ and $d(87)=7$ .) Let $m$ be the number of positive integers $x$ such that $d(x)=20$ . Find the sum of the distinct prime factors of $m$ .
Least number of factors giving 51/2010 ●●●●○ 100 ptsFind the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots, s_n\}$ consisting of $n$ distinct positive integers such that $\left(1-\frac{1}{s_1}\right)\left(1-\frac{1}{s_2}\right)\cdots\left(1-\frac{1}{s_n}\right)=\frac{51}{2010}.$
Denominators of partial sums ●●●●○ 100 ptsOlya wrote fractions of the form $\frac{1}{n}$ on cards, where $n$ runs over all positive divisors of $6^{100}$ (including $1$ and $6^{100}$ ). She arranged the cards in some order. She then wrote on the board the number on the first card, the sum of the numbers on the first two cards, the sum of the numbers on the first three cards, and so on, up to the sum of the numbers on all the cards. Each sum was written as an irreducible fraction. What is the smallest possible number of different denominators that could appear on the board?
Minimum density of a special set ●●●●● 200 ptsLet $S$ be a set of positive integers satisfying the following two conditions: (i) for each positive integer $n$ , at least one of $n, 2n, \ldots, 100n$ is in $S$ ; (ii) if $a_1, a_2, b_1, b_2$ are positive integers such that $\gcd(a_1a_2, b_1b_2)=1$ and $a_1b_1, a_2b_2 \in S$ , then $a_2b_1, a_1b_2 \in S$ . Suppose that $S$ has natural density $r$ , meaning $\lim_{n\to\infty}\frac{1}{n}|S\cap\{1,\ldots,n\}| = r$ . Compute the minimum possible value of $\lfloor 10^5 r\rfloor$ .

More in Divisibility

Counting multiples Divisibility by 11 Divisibility by 3, 4 and 6 Divisibility by 7 and 13 Divisibility by 9 and 99 Largest divisors Largest multiple below a bound Missing digits for divisibility Mixed divisibility conditions Multiples with digit conditions