Prove multiplicative inverses in reduced residue system

Asked Dec 02 '19 at 03:20

Active Dec 02 '19 at 03:20

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Since $d$ is relatively prime to $φ(n)$, it has a multiplicative inverse $e$ in the ring of integers modulo $φ(n)$

This quote presumes that every element in a reduced residue system (modulo n) has a multiplicative inverse (modulo n). Could you please help me prove this fact or provide some intuition?

Beginner-friendly please (preferably no abstract algebra).

asked Dec 02 '19 at 03:20

Jorge Barrios

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every element relatively prime to $\varphi(n)$ has a multiplicative inverse modulo $\varphi(n)$; cf. Bezout's identity – J. W. Tanner Dec 02 '19 at 03:24
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$,q := d,$ is coprime to $,a := \phi(n),$ so by the dupe $,q,$ is invertible $\bmod a,,$ i.e. $,d,$ is invertible $\bmod \phi(n),,$ exactly what is claimed. – Bill Dubuque Dec 02 '19 at 03:47

Prove multiplicative inverses in reduced residue system

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