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1. Count positive factors of 1100 ●○○○○ 10 ptsHow many distinct positive factors does $1100$ have?
2. Nested divisor-count function ●○○○○ 10 ptsLet $\boxed{N}$ mean the number of positive whole number divisors of $N$. For example, $\boxed{3}=2$, because $3$ has two divisors, $1$ and $3$. Find the value of $\boxed{\boxed{11}\times\boxed{20}}$.
3. Common divisors of 48 and 156 ●○○○○ 10 ptsHow many positive divisors do 48 and 156 have in common?
4. Greatest shared divisor of 90 and m ●●○○○ 20 ptsTwo numbers $90$ and $m$ share exactly three positive divisors. What is the greatest of these three common divisors?
5. Divisors of exactly one number ●●○○○ 20 ptsHow many natural numbers divide exactly one of 1080 and 1800, but not both?
6. Composite factors of 420 ●●○○○ 20 ptsFind the number of composite factors of $420$.
7. Gaussian integer pairs with product 10 ●●●○○ 50 ptsCompute the number of ordered pairs of complex numbers $(u, v)$ such that $uv = 10$ and such that the real and imaginary parts of $u$ and $v$ are integers.
8. 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.