What are the possible positive integer values of \(n\) such that \((n - 1)! + 1\) is a power of \(n\)?

There is an \(n\)-dimensional grid in an \(n\)-dimensional Euclidean space made of all points with integer coordinates of the form \((x_1, x_2, \dots, x_n)\) that satisfy the inequality \(0 \leq x_i \leq a_i\) where \(i \in \{1, 2, \dots, n\}\). Every pair of points in the grid that are a unit distance apart are connected by an edge. Through these edges, how many possible shortest paths are there from the point at \((0, 0, \dots, 0)\) to the point at \((a_1, a_2, \dots, a_n\))?

What are the possible positive integer values of \(n\) such that the largest \(n\)-bit integer is a multiple of \(n\)?

In a random permutation of the first \(n\) natural numbers, what is the probability that \(k\) appears before all the numbers greater than itself where \(1 \le k \le n\)?

A partition of a positive integer \(n\) is a list of positive integers, ordered from largest to smallest, such that the sum of the integers in the sequence is \(n\). Each integer in the list is called a part.

What is the ratio of the number of possible partitions of a positive integer \(n\) to the number of possible partitions of \(2n\) into \(n\) parts?