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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? 
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 04 squared mod 5. 
September 27th, 2013, 05:06 AM  #3 
Math Team 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.

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!

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:
 
September 30th, 2013, 12:33 AM  #6  
Math Team 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:
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?  
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:
(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.)  
September 30th, 2013, 12:58 AM  #8  
Math Team 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:
 
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 04 when you square these and reduce modulo 5. Any number, n, greater than 4 is equal to 04 (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). 
September 30th, 2013, 04:16 AM  #10 
Math Team 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.


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