!!!Maybe still helpful but misguiding question|See answer of Doug M for insights|I mistook remainder for result.
I hope you are doing well. So I did this course on Brilliant about infinities. It is structured like a quiz. One question to come up was:
"Suppose you had a complete finite list of every prime (assuming this is even possible to do). You multiplied all of them together, and added one, getting the number Q. What remainder would there be when Q Q Q is divided by any of the primes?"
The options for an answer were:
- 0
- 1
- 2
- it varies based on the prime
So allegedly the answer is:
"Consider Q before the 1 is added to it: Q−1. Since Q−1 is formed by multiplying all primes together, all primes divide evenly into it! So, going back to Q,that means attempting to divide by a prime will result in a remainder of 1 in every case."
Hear me out (and excuse my terrible LaTex^^): If x=any prime of the product Q-1, and Q-1/x=1 and Q-1/Q-1=1 then this infers that Q-1=x. So the answer implies that the product of a given set of primes is equal to any of its multiplicands ?
What am I missing here ?
Suppose we have a list of prime numbers.
We multiply all the numbers on this list together. This number is divisible by every prime number on our list.
Next, we take the number that is the product of all primes on our list and add one. What we now have is not divisible by any number on our list. The remainder will always be 1.
This means that the number we have created... the product of primes + 1 either is prime (and not on our list) or has prime factors that are not on our list. Either way, no list of prime numbers is complete. Hence the set of prime numbers is infinite.