## how to prove that if a=b(mod n) then ....

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### 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
munchiez
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### 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
Carl Friedrich Gauss —
When a philosopher says something that is true then it is trivial.
When he says something that is not trivial then it is false.

Pierre-Simon Laplace —
[said about Napier's logarithms:] ...by shortening the labours doubled the life of the astronomer.

Henri Poincaré —
Les faits ne parlent pas (Facts do not speak).

mathbalarka
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### Re: how to prove that if a=b(mod n) then ....

mathbalarka wrote:By your problem, a isnt divisible by b.

That doesn't follow. (a, b, n) = (2, 2, 2) is possible, and in that example a is divisible by b.
Pari/GP: this is the program I probably mentioned in my post. Windows users can get it at http://pari.math.u-bordeaux.fr/pub/pari ... -2-6-2.exe

CRGreathouse
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### Re: how to prove that if a=b(mod n) then ....

Yes, but what about my next solution,
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

is it ok?
Carl Friedrich Gauss —
When a philosopher says something that is true then it is trivial.
When he says something that is not trivial then it is false.

Pierre-Simon Laplace —
[said about Napier's logarithms:] ...by shortening the labours doubled the life of the astronomer.

Henri Poincaré —
Les faits ne parlent pas (Facts do not speak).

mathbalarka
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Posts: 3759
Joined: Thu Mar 22, 2012 2:44 pm
Location: India, West Bengal

### 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
munchiez
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Posts: 5
Joined: Mon Apr 23, 2012 7:22 am

### 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
Carl Friedrich Gauss —
When a philosopher says something that is true then it is trivial.
When he says something that is not trivial then it is false.

Pierre-Simon Laplace —
[said about Napier's logarithms:] ...by shortening the labours doubled the life of the astronomer.

Henri Poincaré —
Les faits ne parlent pas (Facts do not speak).

mathbalarka
Super User

Posts: 3759
Joined: Thu Mar 22, 2012 2:44 pm
Location: India, West Bengal

### 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

The Chaz
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### Re: how to prove that if a=b(mod n) then ....

The Chaz wrote:a == b

Carl Friedrich Gauss —
When a philosopher says something that is true then it is trivial.
When he says something that is not trivial then it is false.

Pierre-Simon Laplace —
[said about Napier's logarithms:] ...by shortening the labours doubled the life of the astronomer.

Henri Poincaré —
Les faits ne parlent pas (Facts do not speak).

mathbalarka
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Posts: 3759
Joined: Thu Mar 22, 2012 2:44 pm
Location: India, West Bengal

### Re: how to prove that if a=b(mod n) then ....

mathbalarka wrote:
The Chaz wrote:a == b

== is commonly used to mean $\equiv$ when you don't have access to special characters.
Pari/GP: this is the program I probably mentioned in my post. Windows users can get it at http://pari.math.u-bordeaux.fr/pub/pari ... -2-6-2.exe

CRGreathouse
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Posts: 12570
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Location: UTC -5

### 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?
Carl Friedrich Gauss —
When a philosopher says something that is true then it is trivial.
When he says something that is not trivial then it is false.

Pierre-Simon Laplace —
[said about Napier's logarithms:] ...by shortening the labours doubled the life of the astronomer.

Henri Poincaré —
Les faits ne parlent pas (Facts do not speak).

mathbalarka
Super User

Posts: 3759
Joined: Thu Mar 22, 2012 2:44 pm
Location: India, West Bengal

### Re: how to prove that if a=b(mod n) then ....

Ok, got it its a marvelous solution
Carl Friedrich Gauss —
When a philosopher says something that is true then it is trivial.
When he says something that is not trivial then it is false.

Pierre-Simon Laplace —
[said about Napier's logarithms:] ...by shortening the labours doubled the life of the astronomer.

Henri Poincaré —
Les faits ne parlent pas (Facts do not speak).

mathbalarka
Super User

Posts: 3759
Joined: Thu Mar 22, 2012 2:44 pm
Location: India, West Bengal

### Re: how to prove that if a=b(mod n) then ....

CRGreathouse wrote:...
== is commonly used to mean $\equiv$ when you don't have access to special characters.

... or when you're too lazy for LaTex

The Chaz
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Location: Northwest Arkansas

### Re: how to prove that if a=b(mod n) then ....

The Chaz wrote:
CRGreathouse wrote:...
== is commonly used to mean $\equiv$ when you don't have access to special characters.

... or when you're too lazy for LaTex

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!
Carl Friedrich Gauss —
When a philosopher says something that is true then it is trivial.
When he says something that is not trivial then it is false.

Pierre-Simon Laplace —
[said about Napier's logarithms:] ...by shortening the labours doubled the life of the astronomer.

Henri Poincaré —
Les faits ne parlent pas (Facts do not speak).

mathbalarka
Super User

Posts: 3759
Joined: Thu Mar 22, 2012 2:44 pm
Location: India, West Bengal

### 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)

The Chaz
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Joined: Mon Nov 23, 2009 9:34 pm
Location: Northwest Arkansas

### 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)

Im not talking about me, im talking about the author of this topic, munchiez
Carl Friedrich Gauss —
When a philosopher says something that is true then it is trivial.
When he says something that is not trivial then it is false.

Pierre-Simon Laplace —
[said about Napier's logarithms:] ...by shortening the labours doubled the life of the astronomer.

Henri Poincaré —
Les faits ne parlent pas (Facts do not speak).

mathbalarka
Super User

Posts: 3759
Joined: Thu Mar 22, 2012 2:44 pm
Location: India, West Bengal

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