Hi, I have just started learning the fundamentals of Number Theory and I'm having some trouble wrapping my head around a few steps that were presented in my textbook.
the first question contains a simplification of:
7x = 1(mod 8)
-x = 1(mod 8)
a side note states that this is possible because 7=-1(mod 8)
Now, if 7 were congruent to -1(mod 8), and since -1(mod 8) is congruent to -1 (this is what I gathered from their next step), would this imply that 7 is congruent to -1, and hence the next step simply replaces 7 with -1?
The second question contains the following step of simplification:
I can see that the base has reduced by 14, but I am having trouble understanding why exactly 11^n would give the same remainder as 25^n when divided by 14.
I know that that a^n=b^n (mod 14), as well as the other "rules", but I just can't figure it out.
Thanks in advance.