Counting the number of homomorphisms

Question

From what I understand the number of homomorphisms from $Z_m$ to $Z_n$ is the gcd(m,n)

However, I do not know how to count it this way:

use the Fundamental Theorem on Group Homomorphisms (i.e., find a quotient group of $Z_{12}$, a subgroup of $Z_{16}$, and an isomorphism between them)

Answer 1

Both the groups are given as cyclic groups. In such case there is a unique subgroup and unique quotient group for every order that divides the order of the given cyclic group.

So we need to look for numbers that divide both $m$ and $n$. Now it is known that $d$ is a common divisor of $m$ and $n$ iff $d$ divides their gcd.

Answer 2

I won't give any details here, but I will mention something.

Check out the article "The Number of Homomorphisms from $\mathbb{Z}_m$ into $\mathbb{Z}_n$" by Joseph Gallian and James Van Buskirk, American Mathematical Monthly, 91, year 1984.

The article, which is very short, gives not only the number of group homomorphisms $\mathbb{Z}_m \to \mathbb{Z}_n$, but it also gives the number of ring homomorphisms $\mathbb{Z}_m \to \mathbb{Z}_n$.