
Number Theory Number Theory Math Forum 
 LinkBack  Thread Tools  Display Modes 
April 23rd, 2012, 04:31 AM  #1 
Joined: Apr 2012 Posts: 5 Thanks: 0  how to prove that if a=b(mod n) then ....
hey i'm not really sure how i would go about writing out this answer.. "prove that if a = b (mod n) then a^2 = b^2 (mod n) " as in if a and b are congruent to modulo n .. then a squared and b squared are congruent to modulo n ... thanks munchiez 
April 23rd, 2012, 04:42 AM  #2 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,815 Thanks: 42 Math Focus: Number Theory  Re: how to prove that if a=b(mod n) then ....
By your problem, a isnt divisible by b. let a^2 is divisible by b^2. Then (a/b)^2 = k where k is a integer. If k is a nonrootable number the we have nothing to do, it contradicts. If k is a rootable number, then a/b = p where p is an integer. By our primary assumption, it is impossible. So a^2 isnt divisible by b^2. now by you assumption, a = b + nc where c is any integer. Then squaring both sides, it becomes a^2 = b^2 + n(2bc + nc^2) so, a^2 = b^2 (mod n) Q. E. D 
April 23rd, 2012, 04:51 AM  #3  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 12,862 Thanks: 94  Re: how to prove that if a=b(mod n) then .... Quote:
 
April 23rd, 2012, 04:55 AM  #4  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,815 Thanks: 42 Math Focus: Number Theory  Re: how to prove that if a=b(mod n) then ....
Yes, but what about my next solution, Quote:
 
April 23rd, 2012, 05:13 AM  #5 
Joined: Apr 2012 Posts: 5 Thanks: 0  Re: how to prove that if a=b(mod n) then ....
hey thanks man, i guess my maths suck cause i dont quite get it, but i'll try get my head round it

April 23rd, 2012, 05:41 AM  #6 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,815 Thanks: 42 Math Focus: Number Theory  Re: how to prove that if a=b(mod n) then ....
Ok, then let me try again, but this time i will try to be more descriptive, a = b (mod n) means a  b is divisible by n then a  b = n*c where c is an nonzero integer. then a = b + n*c Now squaring both sides , it becomes a^2 = (b + n*c)^2 By the squaring formula, a^2 = b^2 + 2b*n*c + (n^2)*(c^2) = b^2 + n*(2b*c + (c^2)*n) We know that a,b,c,n are integers. So, 2b*c + (c^2)*n is an integer. It is also nonzero because c?0. then 2b*c + (c^2)*n = p where p is a non zero integer. So, a^2 = b^2 + n*p = b^2 (mod n) Q. E. D 
April 23rd, 2012, 05:52 AM  #7 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 2  Re: how to prove that if a=b(mod n) then ....
a == b a  b == 0 (a  b)(a + b) == 0(a + b) == 0 (a^2  b^2) == 0 a^2 == b^2 
April 23rd, 2012, 06:01 AM  #8  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,815 Thanks: 42 Math Focus: Number Theory  Re: how to prove that if a=b(mod n) then .... Quote:
 
April 23rd, 2012, 06:16 AM  #9  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 12,862 Thanks: 94  Re: how to prove that if a=b(mod n) then .... Quote:
 
April 23rd, 2012, 06:23 AM  #10 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,815 Thanks: 42 Math Focus: Number Theory  Re: how to prove that if a=b(mod n) then ....
Didnt got it either, am i missing something? If a == b then how it is related to the original question?

April 23rd, 2012, 06:29 AM  #11 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,815 Thanks: 42 Math Focus: Number Theory  Re: how to prove that if a=b(mod n) then ....
Ok, got it its a marvelous solution

April 23rd, 2012, 06:41 AM  #12  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 2  Re: how to prove that if a=b(mod n) then .... Quote:
 
April 23rd, 2012, 06:49 AM  #13  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,815 Thanks: 42 Math Focus: Number Theory  Re: how to prove that if a=b(mod n) then .... Quote:
But to understand your method, someone needs a strong concept of congruence and its a bit tough to understand for newly NT readers on this forum!  
April 23rd, 2012, 07:02 AM  #14 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 2  Re: how to prove that if a=b(mod n) then ....
I don't think a *strong* understanding of modular arithmetic is necessary... You just have to know that numbers are congruent means that their difference is (congruent to) zero. Using that 1 piece of information on the hypothesis and conclusion leads to a  b == 0 Some step(s) in between... a^2  b^2 == 0 So there's only one step to take a  b to the difference of squares. Is it really that hard to see? Maybe I'm just above average mathematical genius (American version). (by the way, that "genius" comment is a joke about someone we used to know and love) 
April 23rd, 2012, 07:14 AM  #15  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,815 Thanks: 42 Math Focus: Number Theory  Re: how to prove that if a=b(mod n) then .... Quote:
 

Tags 
abmod, prove 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
prove 7>(2^0.5+5^0.5+11^0.5)  Albert.Teng  Algebra  8  February 7th, 2013 09:14 PM 
Goldbach's conjecture (to prove or not to prove)  octaveous  Number Theory  13  September 23rd, 2010 04:36 AM 
prove:  rose3  Number Theory  1  February 16th, 2010 10:23 AM 
prove prove prove. currently dont know where to post  qweiop90  Algebra  1  July 31st, 2008 06:27 AM 
prove prove prove. currently dont know where to post  qweiop90  New Users  1  January 1st, 1970 12:00 AM 