Combinatorics | Number Theory Answers

At a factory where they made toy cars, they have 4 quality inspectors inspecting the cars as they come off the assembly line.
A batch of toy cars is numbered in ORDER from 1 to 2500 and is sent for inspection with each inspector inspecting 1 part.
- every 5th car gets its screen inspected
- every 7th car gets its handle checked
- every 17th car get its wheels tested
- every 23rd car gets its control panel wiring examined

a. explain why NO car is checked by all 4 inspectors.
b. make a list of the numbers of all the cars which are checked by 3 inspectors
d. find a pair of consecutively-numbered cars which are checked by 2 inspectors

Find the largest 7-digit number which satisfy the two properties below
- No two digits can be repeated
- Each of the number’s digits divides into the number exactly

a. Find three digits which cannot be in this number and explain why

b. What is the highest 7 digit number that we can make? Explain why

Let S be a square region (in the plane) of side length 2 inches. Use Pigeonhole Principle to show that among any nine points in S, there are three which are the vertices of a triangle of area is less than or equal to 1/2 square inch.

Seventeen people correspond by mail with one another, each one with all the rest. In their letters only three topics are discussed. Each pair of correspondents deals with only one of the topics. Use Pigeonhole Principle to prove that there are at least three people who write to each other about the same topic

Solve the equation 3φ(n) =4τ(n) where φ(n) is the Euler totient function and τ(n) is the number of divisons of n.