-3
$\begingroup$

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.)

$\endgroup$

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

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – choco_addicted, Arnaud D., Error 404, rtybase, Davide Giraudo
If this question can be reworded to fit the rules in the help center, please edit the question.

1
$\begingroup$

For

$(a)$ 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!

$\endgroup$
  • $\begingroup$ 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

Not the answer you're looking for? Browse other questions tagged or ask your own question.