How Can a^r = 1 mod N Be Expressed Differently If a and N Are Coprime?

[Baggio]

If a and N coprime, a < N and r is even how can

a^(r) = 1 mod(N) be written as


a^(r) - 1 = 0 mod(N) ?

I don't know much number theory (I'm a physics student)

Thanks

 

[shmoe]

Hi, the conditions you list at the start don't matter for what you've just asked (a, N coprime, etc). For any a,b, and N,

a=b mod (N)

if and only if

N divides a-b

if and only if

N divides (a-b)-0

if and only if

a-b=0 mod(N)


Working mod N you can add, subtract and multiply as normal (division isn't always possible!), you just have the luxury of reducing things mod(N) afterwards.

 

[Baggio]

ahh i see what you did there.. thanks!