This might look like a copy of another question, but what I'm about to propose here is new. There's this question,

> 
> *Find the least positive integral value of n for which $(\frac{1+i}{1-i})^n = 1$*
> 
> 
> 

While solving, if we multiply what is within the bracket, by the conjugate of denominator and divide by the same thing, we get **$i$** in the bracket, that means, the question boils down to

> 
> **$i ^ n= 1$**
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> 
> 

We know that the least positive value $n$ can have is $4$, for $i^n$ to be $1$.
Done.
Now,
IF I were to solve it by taking mod on both sides of the given equation,
I would get

> 
> **$\Big(\frac{|1+i|}{|1-i|}\Big)^n = |1|$**
> 
> 
> **$\Big(\frac{\sqrt{2}}{\sqrt{2}}\Big)^n = 1$**
> 
> 
> **$1^n = 1$**
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> 
> 

NOTE that the least positive value of $n$ changes from $4$ to $1$.

Why is it so?
I read an answer on stack exchange, that it is valid to do anything with the equation until it maintains the equality. I didn't destroy the equality, so why does the answer vary?

Is there any restriction as to where to use the "taking-mod-both-sides" thing?