|
|
|
|Store|
|
|
|
|
|
|
|
|
CTK Exchange
rossnoe
Member since Mar-3-04
|
Mar-03-04, 06:56 PM (EST) |
    |
"Composite question"
| |
I'm taking a number theory course to complete my undergraduate math degree. I want to know if my approach to solving this problem is correct: -----------------------------------------------------------
Question:
Show that if n is composite then there exists a prime p less than or equal to the square root of n such that p|n.(Hint: Consider what happens when 2 numbers greater than are multiplied).
-----------------------------------------------------------
Answer:
p|n means that n/p = c and n = cp. c is some integer. p less than or equal to the square root of n. therefore (p)^2 less than or equal to (the square root of n)^2 , hence p^2 less than or equal n, p^2 less than or equal cp, p less than or equal c In the case where p = c
n = cp leads to n = c^2 and c = the square root of n. As stated above c is an integer so n is a perfect square.
----------------------------------------------------------- |
|
|
Alert | IP |
Printer-friendly page |
Reply |
Reply With Quote | Top |
|
|
 |
|
|
alexb
Charter Member
1229 posts |
Mar-08-04, 10:55 AM (EST) |
|
|
6. "RE: Composite question"
In response to message #4
| |
>Hmmmm. Can you be a bit more specific? What doesn't make
>sense?
Doing things in reverse.You ask "How do we know that p ≤ √n?" Then you start a derivation "p ≤ √nn. Therefore, ..." You make a derivation from something you have just asked about whether it's true or not. This does not make sense. "p ≤ √nn" is something you want to prove.
|
|
|
Alert | IP |
Printer-friendly page |
Reply |
Reply With Quote | Top |
|
|
|
|
Ralph Boles
guest
|
Mar-14-04, 06:54 AM (EST) |
|
|
8. "RE: Composite question"
In response to message #7
| |
This is not hard. First assume that the statement is false i.e. that the prime factorization of n contains only primes larger than sqrt(n) and think about what that implies for n. Use the hint. |
|
|
Alert | IP |
Printer-friendly page |
Reply |
Reply With Quote | Top |
|
|
|
You may be curious to have a look at the old CTK Exchange archive.
Please do not post there.
|Front page|
|Contents|
Copyright © 1996-2018 Alexander Bogomolny
71546410
|
|
|
Printer-friendly copy
Email this topic to a friend
