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Will all units in $\mathbb{Z}_{72}$ be also units (modulo $8$ and $9$) of $\mathbb{Z}_8$ and $\mathbb{Z}_9$?

I think yes, because if $\gcd(x,72)=1\implies\gcd(x,8)=\gcd(x,9)=1$, right? any counterexamples? Thanks beforehand.

Simon Fraser
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vidyarthi
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    As a matter of pedantry, $\Bbb Z_{72}$ and $\Bbb Z_8$ are disjoint. They have no elements in common. But you can compare the set of integers whose congruence classes in $\Bbb Z_{72}$ are units to the set of integers whose congruence classes in $\Bbb Z_8$ are units. – Arthur Jan 01 '20 at 22:13
  • The group of units modulo $n$ is $$U(n)\cong {\rm Aut}(\Bbb Z_n).$$ – Shaun Jan 01 '20 at 22:17

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Yes, that is correct. More generally if $\,f(x)\,$ is polynomial with integer coef's and it has a root $\,x\equiv r\pmod{\!mn}\,$ then that root persists $\bmod m\,$ & $\,n\,$ since congruences persist mod factors of the modulus (or directly: $\,m,n\mid mn\mid f(r)).\,$ Specializing $\,f(x) = ax-1\,$ shows that if $\,a\,$ is a unit $\bmod mn$ then it persists as a unit $\!\bmod m\,$ & $\,n.$

Or, up in $\Bbb Z,\,$ we can view it as persistence of Bezout identities for $\rm\color{#c00}{factors}$ of gcd arguments, i.e.

$$\gcd(\color{#c00}ab,\color{#c00}mn)=1\,\Rightarrow\, abx + mny = 1\,\Rightarrow a(bx)+m(ny) = 1\,\Rightarrow\, \gcd(\color{#c00}{a,m}) = 1$$

Or, in $ $ divisibility $ $ language: $\ \ (a,m)\ \mid\ (ab,mn) = 1$

Or, in simpler ideal language $\ (a,m)\supseteq (ab,mn) = (1)$

Bill Dubuque
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A bit more abstractly: If you have a homomorphism $\varphi: R \to S$ of unital commutative rings, then for any unit $r \in R$, its image $\varphi(r)\in S$ is a unit, because $\varphi(r) \varphi(r^{-1}) = \varphi(r r^{-1}) = \varphi(1) = 1$.

In your situation, you have surjections $\mathbb{Z} \to \mathbb{Z}_{72} \to \mathbb{Z}_8$, so any number that is a unit mod 72 will also be a unit mod 8.

red_trumpet
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    Worth mention: that argument is in fact a special case of the persistence of polynomial roots - as I mentioned (I used congruences vs. quotient rings since it wasn't clear the OP knew them). – Bill Dubuque Jan 01 '20 at 23:29