Integers modulo 7 follow the inverse property of multiplication, but integers modulo 9 do not; is there a reason why?
2026-03-25 17:44:52.1774460692
Why do some integer modulos have no inverse?
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If you have a prime number $p$ for your modulo,then you always get an inverse for $\{1,2,\cdots,p-1\}$. And then this set is called finite field, meaning on this set addition, subtraction, multiplication, and division are defined.
https://en.wikipedia.org/wiki/Finite_field
However, if you do not have a prime, then there are always elements without the inverse. I guess you are asking why this is.
For that, the reason is well written here: https://en.wikipedia.org/wiki/Modular_multiplicative_inverse Only prime numbers are relatively prime with all the numbers smaller than the original prime.