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15 problems match “prime factorisation”. Clear

Divisors giving equal quotient and remainder ●●●○○ 50 ptsWhen 2014 is divided by 105, the quotient is 19 and the remainder is also 19. Find all natural numbers $N$ other than 105 such that dividing 2014 by $N$ gives a quotient equal to the remainder. List the values in increasing order.
Terminating decimals with nonzero hundredths digit ●●●○○ 50 ptsFor how many positive integers $n$ does $\frac{1}{n}$ yield a terminating decimal with a non-zero hundredths digit?
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 $m$ .
Minimal f(2007) for a functional equation ●●●●○ 100 ptsLet $f$ be a function taking the positive integers to the positive integers, such that $f(mf(n)) = nf(m)$ for all positive integers $m$ and $n$ . Find the smallest possible value of $f(2007)$ .
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?