What is the maximal order among elements in $\mathbb{Z}\times(\mathbb{Z}/10\Bbb{Z})/\langle(5,4)\rangle$?

Question

Here's a problem about a group : $\mathbb{Z}\times(\mathbb{Z}/10\Bbb{Z})$

Problem : Find the maximal order of elements of $\mathbb{Z}\times(\mathbb{Z}/10\Bbb{Z})/\langle(5,4)\rangle$

The answer is...

$(\mathbb{Z}\times(\mathbb{Z}/10\Bbb{Z}))/\langle(5,4)\rangle \cong(\mathbb{Z}\times\mathbb{Z})/\langle(0,10),(5,4)\rangle \cong(\mathbb{Z}\times\mathbb{Z})/\langle(0,1),(50,0)\rangle \cong\mathbb{Z}/50\Bbb{Z}$. So, maximal order is $50$.

I can't understand why $(\mathbb{Z}\times\mathbb{Z})/\langle(0,10),(5,4)\rangle\cong(\mathbb{Z}\times\mathbb{Z})/\langle(0,1),(50,0)\rangle$.

Could you explain why it is? Thank you in advance.

Answer 1

The exercise is wrong. The exercise seems to imply that $$\Bbb{Z}\times(\Bbb{Z}/10\Bbb{Z}) \cong(\Bbb{Z}\times\Bbb{Z})/\langle(0,1)\rangle,$$ which is false. Instead this should be $$\Bbb{Z}\times(\Bbb{Z}/10\Bbb{Z}) \cong(\Bbb{Z}\times\Bbb{Z})/\langle(0,10)\rangle,$$ which you should prove for yourself, if this is not yet clear. Then accordingly, the first isomorphism should be \begin{eqnarray*} \Bbb{Z}\times(\Bbb{Z}/10\Bbb{Z}))/\langle(5,4)\rangle &\cong&(\Bbb{Z}\times\Bbb{Z})/\langle(0,10),(5,4)\rangle.\\ \end{eqnarray*} Can you continue from here?