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

  • Thread starter Baggio
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In summary, the conversation discusses the relationship between a^(r) = 1 mod(N) and a^(r) - 1 = 0 mod(N) and how working mod N allows for the manipulation of equations using addition, subtraction, and multiplication. The initial conditions of a and N being coprime and r being even are not relevant to the question.
  • #1
Baggio
211
1
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
 
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  • #2
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.
 
  • #3
ahh i see what you did there.. thanks!
 

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

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