Counting Divisors
For any positive integer \(x\), let \(\tau(x)\) denote the number of positive integer factors of \(x\) Including \(1\) and \(x\) itself. For example \(\tau(3)=2\) and \(\tau(12)=6\). Given two numbers \(X\) and \(Y\), your task is to find the smallest positive integer \(N\) such that \(\tau(N+1)=X\) and \(\tau(N-1)=Y\).
| Part A | \(X = 18\) and \(Y = 8\) | |
|---|---|---|
| Part B | \(X = 108\) and \(Y = 3\) |
Nested Radicals
Consider the following expressions.
\( A = \sqrt{2\sqrt{3\sqrt{4\sqrt{5\sqrt{6 \ldots }}}}} \)
\( B = \sqrt{-1 + 3\sqrt{-2 + 5\sqrt{-3 + 7\sqrt{-4 + 9\sqrt{-5 + \cdots }}}}} \)
| Part A | What are the \(8\) digits of \(A\) immediately following the decimal point? | |
|---|---|---|
| Part B | What are the \(8\) digits of \(B\) immediately following the decimal point? |