For a positive integer n, define n factorial to be the integer n! = n(n − 1)(n − 2)· · · 1.

(a) Suppose 1 ≤ k ≤ n. What are the quotient and remainder when N = n! + 1 is divided by k? Explain.

(b) Explain why part (a) implies that N has a prime divisor greater than n.

(c) Explain why part (b) implies that there are infinitely many prime numbers. (Note: if there are only finitely many prime numbers, then there is a largest prime.)


closed as off-topic by choco_addicted, Arnaud D., Error 404, rtybase, Davide Giraudo Oct 19 '17 at 15:25

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$(a)$ think what you get as rest if you divide $N$ with $k$. Since $k$ is a factor of $N-1$ for all $k$. What happens with that $1$?

$(b)$ think like this: by the fundamental theorem of of arithmetic an arbitrary number $m\in\mathbb{N}^+$ is either a prime or it is a composite. So $N$ is either a prime or composite. Since all the numbers up to $n$ cannot divide $N$ by $(a)$ what happens with $N$s unique factorisation?

$(c)$ for this imagine that we have a finite number of primes. Multiply them together and add one to it and than look at the result from $(b)$.

Hope this helps. If you need more guidance just let me know :)

Good luck!

  • $\begingroup$ thanks that helped alot $\endgroup$ – UnXnown Oct 19 '17 at 9:04
  • $\begingroup$ welcome. when you are done with the solution, add an own answer or accept mine so we have the question answered :) $\endgroup$ – Vinyl_cape_jawa Oct 19 '17 at 9:07
  • $\begingroup$ and then ppl we remove the downvotes from the question as well I think $\endgroup$ – Vinyl_cape_jawa Oct 19 '17 at 9:08

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