Is the book wrong about this corollary? My book states

The group $\prod_{i = 1\dots n} \mathbb{Z}_{m_i}$ is cyclic and isomorphic to $\mathbb{Z}_{m_1m_2\dots m_n} \iff$ the numbers $m_i$ for $i = 1\dots n$ are such that the gcd of any two of them is $1$.

So if I take $\mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_{10}$, then gcd(2,3) = 1, so $\mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_{10} \equiv \mathbb{Z}_6 \times \mathbb{Z}_{10}$, but gcd(6,10) $\neq 1$, so $\mathbb{Z}_6 \times \mathbb{Z}_{10}$ is not isomorphic to $\mathbb{Z}_{60}$, so this is wrong?
 A: I think this wording is extremely unclear.  "If any two members of a set S satisfy condition P, then Q" parses (in my mind) as "If there exists a pair of elements in S that satisfy P, then Q".  But the authors intend it to mean "If all pairs of elements in S satisfy P, then Q."  I think that is the essence of your question.
A: You are correct.  For $\mathbb{Z}_{m_1} \times \mathbb{Z}_{m_2} \times ... \times \mathbb{Z}_{m_n}$ to be isomorphic to $\mathbb{Z}_{m_1m_2...m_n}$, then the $m_i$'s must be pairwise coprime.  In other words, $\gcd(m_i, m_j) = 1$ for all $m_i$, $m_j$ when $i \neq j$.  
So, for example:
$$\mathbb{Z}_3 \times \mathbb{Z}_4 \times \mathbb{Z}_5 \cong \mathbb{Z}_{60}$$
But:
$$\mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_6 \cong \mathbb{Z}_6 \times \mathbb{Z}_6 \ncong \mathbb{Z}_{36}$$
Here are some more examples:


*

*If the $m_i$'s are a collection of distinct primes, then the product
of the cyclic groups will be cyclic.  

*If two of the $m_i$'s are even, then the product will not be cyclic. 

*If $m_i = m_j$ for some $i \neq j$, then the product will not be
cyclic.
