How many zero divisors at $\mathbb{Z}_{80}\times\mathbb{Z}_{100}$?

Question

How many zero divisors at $\mathbb{Z}_{80}\times\mathbb{Z}_{100}$?
I assume it: $(80-\varphi(80))\cdot(100-\varphi(100))$,
I'm right or I miss somthing??

Thank you!

Answer 1

$(a,b)\in A\times B$ is a zero divisor iff $a$ is a zero divisor in $A$ or $b$ is a zero divisor in $B$. Note that e.g. $(1,0)$ is a zero divisor because $(1,0)\cdot(0,1)=(0,0)$.

So you should have $80\cdot 100-\phi(80)\cdot \phi(100)$ instead.