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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: 3,871 Thanks: 85 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: 15,035 Thanks: 664 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic  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: 3,871 Thanks: 85 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: 3,871 Thanks: 85 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: 3  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: 3,871 Thanks: 85 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: 15,035 Thanks: 664 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic  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: 3,871 Thanks: 85 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?


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