You are given a math word problem and you are supposed to make use of mathematical concepts like probability, combinatorics, linear equations, sequences/series, or simple operators like addition, subtraction, division, or multiplication to answer the given question. Choose the best option from the 5 given options and report that option.

[Q]: Question: find two integers, neither of which ends in a zero, and whose product is exactly 10,000

 Option A: 64 and 15,625

 Option B: 60 and 15,625

 Option C: 64 and 15,620

 Option D: 16 and 625

 Option E: 64 and 16,625
[A]: Option D


[Q]: Question: A certain vehicle is undergoing performance adjustments during a set of three trials on an 8-mile track. The first time it travels around the track at a constant rate of p miles per minute, the second time at a constant rate of p^2 miles per minute, and the third time at a constant rate of p^3 miles per minute. If the vehicle takes the same time to travel the first 18 miles of these trials as it does to travel the last 8 miles, how many minutes does it take to complete all three trials (not counting any time between trials)?

 Option A: 18

 Option B: 36

 Option C: 60

 Option D: 96

 Option E: 112 minutes
[A]: Option E


[Q]: Question: Let n~ be defined for all positive integers n as the remainder when (n - 1)! is divided by n.
What is the value of 34~ ?

 Option A: 6

 Option B: 1

 Option C: 2

 Option D: 0

 Option E: 31
[A]:
Option D