# Property of Modular arithmetic

If I know that $$g^a \neq 1 \mod b$$ is that always true that if I will take a positive integer $c$ and count $(g^a)^c$, then $$(g^a)^c \neq 1 \mod b$$?

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No. It's never true when $g$ is a unit mod $b$, e.g. take $c=\varphi(b)$. –  blue May 20 '14 at 18:27
So if $c = \phi(b)$ it is always true? –  Rop May 20 '14 at 18:31
The relation $x^{\varphi(b)}\equiv1$ mod $b$ is true for any unit $x$. In particular if $g$ is a unit then $g^a$ is a unit for any $a\in\Bbb Z$. –  blue May 20 '14 at 18:32
@seaturtles I understand, thank you –  Rop May 20 '14 at 18:34
@seaturtles If I can ask you one more question: if I know that $g^{\phi(b)/x} \neq 1 \mod b$, can I also say that always $(g^{\phi(b)/x})^c \neq 1 \mod b$? –  Rop May 20 '14 at 18:50

Hint $\$ In a finite commutative ring every element is a unit or a zero-divisor. By Lagrange's theorem, a unit has finite order (and the converse is clear),  so the units are precisely the elements of finite order, so the elements you seek, those not of finite order, are precisely the zero-divisors.
Iff $\gcd(g^a,b) > 1$. Otherwise, take $c = \phi(b)$.