Relatively prime numbers and Pi

A prime number is a number whose only factors are 1 and itself. Examples are 2, 3, 5, 7, 11, 13, and so on. Two numbers are relatively prime if they have no common factors aside from 1. For example, 12 and 35 are relatively prime since 12 = 2*2*3 and 35 = 5*7 have no common factors.

The chance that two randomly chosen numbers are relatively prime is 6/Pi^2 or approximately 60%.

To test this, I let 10 students write two numbers. The result is close. Out of 10 pairs, 7 pairs are relatively prime. That is 70%. If there are 100 random pairs, the result would be closer.

Game 23

Here's a trick that will surely amaze students who have not yet discovered the real power of algebra:

Face the students with the blackboard behind you. Ask for one volunteer to write a number between 50 and 100 on the board. Ask him/her to add 76 to the number. Next, ask him/her to add the first digit of the result to the last two-digit number. Finally, subtract the result from the original number he wrote on the board. Surprise the entire class by "guessing" that the final number on the board is 23.

Example:
original number = 53
add 76 to 53 = 129
add 1 to 29 = 30
subtract 30 from 53 = 23

* I discovered recently that there is a coming movie with the title "Game 23".

Goldbach's counterexample?

Find a composite even number that cannot be expressed as a sum of two prime numbers.