My Math Forum  

Go Back   My Math Forum > College Math Forum > Number Theory

Number Theory Number Theory Math Forum


Reply
 
LinkBack Thread Tools Display Modes
September 27th, 2013, 02:40 AM   #1
Newbie
 
Joined: Sep 2013

Posts: 6
Thanks: 0

n^2 divided by 5 will always leave remainder 1 or 4, why?

Hi. Hoping for some help.

When n is an integer n>/=3 and you divide n^2 by 5, the remainder will always be 1 or 4, correct? I need to understand why, but I can't seem to figure it out. Any help out there?
getwolfgang is offline  
 
September 27th, 2013, 02:50 AM   #2
Senior Member
 
Joined: Jun 2013
From: London, England

Posts: 1,316
Thanks: 116

Re: n^2 divided by 5 will always leave remainder 1 or 4, why

Or 0.

You just need to consider 0-4 squared mod 5.
Pero is offline  
September 27th, 2013, 05:06 AM   #3
Math Team
 
mathbalarka's Avatar
 
Joined: Mar 2012
From: India, West Bengal

Posts: 3,871
Thanks: 86

Math Focus: Number Theory
Re: n^2 divided by 5 will always leave remainder 1 or 4, why

Basically, what you need to understand is the n divided by 5 can leave reminders 0, 1, 2, 3 or 4. The only squares are 0 and 4, so clearly, 0^2 and 4^2 divided by 5 leaves reminder 0 and 1, respectively.
mathbalarka is offline  
September 30th, 2013, 12:06 AM   #4
Newbie
 
Joined: Sep 2013

Posts: 6
Thanks: 0

Re: n^2 divided by 5 will always leave remainder 1 or 4, why

Ah, ofc. Thank you!
getwolfgang is offline  
September 30th, 2013, 12:10 AM   #5
Newbie
 
Joined: Sep 2013

Posts: 6
Thanks: 0

Re: n^2 divided by 5 will always leave remainder 1 or 4, why

Quote:
Originally Posted by mathbalarka
Basically, what you need to understand is the n divided by 5 can leave reminders 0, 1, 2, 3 or 4. The only squares are 0 and 4, so clearly, 0^2 and 4^2 divided by 5 leaves reminder 0 and 1, respectively.
But wait, now I'm confused again. What about the reminder 4...?
getwolfgang is offline  
September 30th, 2013, 12:33 AM   #6
Math Team
 
mathbalarka's Avatar
 
Joined: Mar 2012
From: India, West Bengal

Posts: 3,871
Thanks: 86

Math Focus: Number Theory
Re: n^2 divided by 5 will always leave remainder 1 or 4, why

Quote:
Originally Posted by getwolfgang
But wait, now I'm confused again. What about the reminder 4...?
Yes. If n^2 is divisible by 5, then the reminder left after dividing n by 5 can be 0 or 4, since they are the only squares in the list {0, 1, 2, 3, 4}.

So, we have : reminder after dividing n by 5 is 0 or 4
Hence : reminder after dividing n^2 by 5 is 0^2(=0) or 4^2 (=1)

Get it?
mathbalarka is offline  
September 30th, 2013, 12:52 AM   #7
Newbie
 
Joined: Sep 2013

Posts: 6
Thanks: 0

Re: n^2 divided by 5 will always leave remainder 1 or 4, why

Quote:
Originally Posted by mathbalarka
Quote:
Originally Posted by getwolfgang
But wait, now I'm confused again. What about the reminder 4...?
Yes. If n^2 is divisible by 5, then the reminder left after dividing n by 5 can be 0 or 4, since they are the only squares in the list {0, 1, 2, 3, 4}.

So, we have : reminder after dividing n by 5 is 0 or 4
Hence : reminder after dividing n^2 by 5 is 0^2(=0) or 4^2 (=1)

Get it?
Isn't 1 a square? (1^2 = 1) and would give the reminder 4, so the answer would be that the possible reminders are 0, 1 and 4?
(If I'm completely misunderstanding you, then I appologize, I'm really tired and my brain isn't working as fast as I would like.)
getwolfgang is offline  
September 30th, 2013, 12:58 AM   #8
Math Team
 
mathbalarka's Avatar
 
Joined: Mar 2012
From: India, West Bengal

Posts: 3,871
Thanks: 86

Math Focus: Number Theory
Re: n^2 divided by 5 will always leave remainder 1 or 4, why

Quote:
Originally Posted by getwolfgang
Isn't 1 a square? (1^2 = 1) and would give the reminder 4, so the answer would be that the possible reminders are 0, 1 and 4?
Ah, right. n divided by 5 leaves reminder 0, 1 and 4, but that wouldn't change the answer would it? Squaring the reminders gives 0 and 1.
mathbalarka is offline  
September 30th, 2013, 01:15 AM   #9
Senior Member
 
Joined: Jun 2013
From: London, England

Posts: 1,316
Thanks: 116

Re: n^2 divided by 5 will always leave remainder 1 or 4, why

The critical thing is what happens to the numbers 0-4 when you square these and reduce modulo 5. Any number, n, greater than 4 is equal to 0-4 (mod 5) and n^2 (mod 5) is the same as (n (mod 5))^2 (mod 5).

So:

0^2 = 0 (mod 5)
1^2 = 1 (mod 5)
2^2 = 4 (mod 5)
3^2 = 4 (mod 5)
4^2 = 1 (mod 5)

So, any number squared will be 0, 1 or 4 (mod 5).
Pero is offline  
September 30th, 2013, 04:16 AM   #10
Math Team
 
mathbalarka's Avatar
 
Joined: Mar 2012
From: India, West Bengal

Posts: 3,871
Thanks: 86

Math Focus: Number Theory
Re: n^2 divided by 5 will always leave remainder 1 or 4, why

If OP had understood modular arithmetic he wouldn't have asked this question at all.
mathbalarka is offline  
Reply

  My Math Forum > College Math Forum > Number Theory

Tags
divided, leave, remainder



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
To leave a film that has 4 doors itamaratento Elementary Math 7 January 31st, 2014 03:01 AM
Transformations that leave invariant a binomial distribution becko Applied Math 0 August 18th, 2011 08:22 PM
14 divided by 3? helllol Algebra 2 September 22nd, 2010 03:47 PM
Find the remainder when 16! Is divided by 17 . gigglie Number Theory 4 May 3rd, 2008 10:02 AM
To leave a film that has 4 doors itamaratento Abstract Algebra 1 December 31st, 1969 04:00 PM





Copyright © 2019 My Math Forum. All rights reserved.