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1)Definition: Let m denote a positive integer and a any integer such that gcd(a,m)=1. Let h be the smallest positive integer such that a^{h}≡ 1 (mod m). Then h is called theorderof a modulo m. (notation: h=e_{m}(a) )

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Now, why do we need to assume gcd(a,m)=1 in this definition of order? Is it true that if gcd(a,m)≠1, then the order e_{m}(a) is undefined? Why or why not? I can't figure this out. I know that the multiplicative inverse of a mod m exists <=> gcd(a,m)=1, but I don't see the connection...

2) Definition: For all a such that gcd(a,m)=1, a is called aquadratic residuemodulo m if the congruence x^{2}≡ a (mod m) has a solution. If it has no solution, then a is called a quadratic nonresidue modulo m.

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Once again, why is the assumption gcd(a,m)=1 needed? What happens when gcd(a,m)≠1?

May someone explain, please?

Thanks a million!

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# Order of a mod m & quadratic residues

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