let m be a positive integer and let a be an integer with (a,m)=1. (a) Prove that if Ordm(a)=xy (with x and y positive integers), then ordm(a^x)=y (b) prove that if ordm(a)=m-1, then m is a prime number.
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Originally Posted by mndi1105 let m be a positive integer and let a be an integer with (a,m)=1. (a) Prove that if Ordm(a)=xy (with x and y positive integers), then ordm(a^x)=y Because . Now if so that then it means and but is order of . This is a contradiction. Thus the order of must be (b) prove that if ordm(a)=m-1, then m is a prime number. That is because . The inequalities force . This happens if and only if is prime.
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